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Full text of "Littoral sand transport on beaches"

Copyright 19 7 9 Todd L. Walton Jr. Lif^oiAL Bmj3 immsmm ow benches TQDD L. WALTON JR. A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA ACKNOWLEDGEMENTS With utmost sincerity, the author wishes to thank Dr. B.A. Christensen and Dr. T. Y. Chiu for their advice, direc- tion, and considerable assistance which they have given to the author's Ph.D. program. Without their untiring help and encouragement this thesis would not have been possible. Thanks are also due to Dr. D. P. Spangler, Dr. F. W. Morris, and Dr. J. G. Melville, for serving as the author's supervisory committee. The author also wishes to express his deep appreciation to his former mentor, Dr. R. G. Dean whose efforts in helping the author to understand the basics of science have never ceased, and to his good friend the late Professor J. A. Purpura whose encouragement and personal understanding of shore processes were of tremendous benefit to the author. Appreciation is also extended 'to the Florida Sea Grant Program whose grant C0M--3-158-43 ) made many of the early results of this thesis possible, and the Department of Natural Resources, Bureau of Beaches and Shores, State of Florida, whose support and encouragement have benefitted the author . Finally, the author wishes to thank his wife Cindy whose patience and unselfishness throughout the pursuit of this dissertation has made life more wonderful. iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS iii LIST OF TABLES vii LIST OF FIGURES ix LIST OF SYMBOLS xvi ABSTRACT xxiv CHAPTER I. INTRODUCTION 1 1-1 Introductory Note 1 1.2 Statement of Erosion Problem 3 1.3 Objectives 17 II. REVIEW OF SURF ZONE MODELS 19 2.1 Review of Existing Longshore Sand Transport Models 19 2 . 2 Review of Existing Longshore Current Models 2 7 2.3 Review of Friction Factor 39 2.3.1 Field Data i+0 2.3.2 Laboratory Data i44 III. REVIEW OF EXISTING DATA ON LONGSHORE SAND TRANSPORT 51 3.1 Introductory Remarks 51 3.2 Laboratory Studies 51 3.2.1 Krumbein Laboratory Study 53 3.2.2 Seville Laboratory Study 5 6 3.3.3 Shay and Johnson Laboratory Study . 56 3 . 3 . M- Sauvage and Vincent Laboratory Study 5 8 3.3.5 Savage and Fairchild Laboratory Study 5 8 3.3.6 Price, Tomlinson, and Willis Laboratory Study 6 0 3.2.7 Barcelo Laboratory Study 61 3.2.8 Bijker Laboratory Study 61 IV Page 3.3 Field Studies 62 3.3.1 Watt Field Study 5 5 3.3.2 Caldwell Field Study 69 3.3.3 CERC Field Study 70 3.3.4 Moore and Cole Field Study .... 70 3.3.5 Komar Field Study 71 3.3.6 Thornton Study 7 2 IV. SAND TRANSPORT MODEL DEVELOPMENT 7 3 4-.1 Introduction 7 3 M- . 2 Review of Mass, Momentum, and Energy Conservation in Periodic Wave Flow ... 76 4.3 Review of Wave Field Equations 87 4.14 Proposed Sand Transport Model 105 4.4.1 Longshore Current Estimation . , .10 5 4.4.2 Integrated Sand Transport Estimation 120 4.4.3 Sand Transport Distribution Across Surf Zone 131 4.4.4 Estimation of Efficiency Factors and Dynamic Friction Coefficient 13 8 4.4.5 Model Comparison with Laboratory and Field Results 155 V. APPLICATION OF SAND TRANSPORT EQUATION TO CALCULATION OF LITTORAL DRIFT USING SHIP WAVE DATA 16 2 5.1 Application of Model 16 2 5.1.1 Data Source' 16 4 5.1.2 Analysis of Wave Data to Compute Sand Transport 16 5 5.2 Results of Sand Transport Computations 17 2 5.2.1 Use of a Littoral Drift Rose . . . 175 5.2.2 Possible Sources of Data Error or Bias 178 5.2.3 Other Possible Errors or Bias in Analysis 180 5.2.4 Other Potential Sources of Error 182 5 . 3 Comparison of Calculated Littoral Drift Rates with Previously Estimated Values 185 5.4 Comparison of Estimated and Observed Wave Climates 19 8 VI. ANALYTICAL MODELS FOR SHORELINE CHANGE . . . 20 8 6.1 Introductory Rem^arks 208 V Page 6 . 2 Heuristic "Equilibrium" Shoreline Model Development 209 6 . 3 , Analytical Treatment of Shoreline Change Model 229 VII. SAND TRANSPORT AND STORAGE AT INLETS .... 248 VIII. SUMMARY AND CONCLUSIONS 259 REFERENCES 262 APPENDIX A. DERIVATION OF LONGSHORE ENERGY FLUX PER UNIT LENGTH SHORELINE 275 B. LONGSHORE SAND TRANSPORT MODEL DATA SUSPENDED SAND CONCENTRATION DATA 2 80 C. ANALYSIS OF SSMO WAVE HEIGHT, PERIOD, AND DIRECTION RANGES 289 BIOGRAPHICAL SKETCH . ' ■ . 321 vi LIST OF TABLES Table Page 1.1 Volumes of Material Present in Outer Inlet Shoals of Florida Inlets . 11 1.2 Beach Nourishment Projects 12 2.1 Littoral - Sand Transport, .Models 20 2.2 Littoral Sand Transport Models of Equation Type 2.1 23 2.3 Longshore Current Formulas 29 2.4- Results of Field Measurements of Fric- tion Factor i43 2.5 Results of Laboratory Measurements of Friction Factor i+8 3.1 Testing Variables of Laboratory Data 54 3.2 Summary of Wave Basin Dimensions 55 3- 3 Field Study Tests for Sand Transport Determination 56 4- .1 Observed and Theoretical Values of 4.2 Integrals for Sediment Transport 137 5.1 Comparison of Annual Net Littoral Drift Rates as Estimated by the U.S. Army Corps of Engineers and as Calculated in the Present Study 183 5.2 Comparison of Annual Average Total Lit- toral Drift Rates as Estimated by the U.S. Army Corps of Engineers and as Calculated in the Present Study 187 vi.i LIST OF TABLES (Continued) Table Page 5.3 Recording Periods of Shore Based CERC Wave Gages Used in Comparison of Actual to Predicted Shore Wave Climate 200 B.l Shay and Johnson Data 280 B.2 Komar Data 281 B.3 Fairchild Data 282 B.4 Watts Data 283 B.5 Kana Data 2B^■ B. 6 Fairchild Data 285 C. l Representative Values of Wave Height Used in Computation of Longshore Energy Flux 291 C.2 Representative Values of Wave Period Used in Computation of Longshore Energy Flux 292 ::v ill LIST OF FIGURES -^^g^^ Page 1.1 Erosion Situation in Florida 2 1.2 Relative Rise in Sea Level in Florida t| 1.3 Bruun's Concept of Beach Profile Response to Sea Level Rise 5 1.4 Sediments Deposited off Central Padre Island During and After Hurricane Carla on a Mud Bottom 7 1.5 General Beach-Dune Profile Before and After Hurricane Eloise Between Panama City and Destin, Florida 9 1.5 Deposition of Sand in the Interior of St. Lucie Inlet 14. 1.7 Relationships Between Outer Bar Stor- age and Cross Sectional Area of Inlets 15 3.1 Distribution of Longshore Velocity and Sediment Transport Ac:toss the Surf Zone , 61+ 14.1 Shear Stresses Acting on the Faces of a Water Column 84 4.2 Schematic of the Sur;' Zone 92 4.3 Schematic Plan and Section of the Near- shore Region 108 4.4 Shear Forces Due to Oscillatory Wave Velocity and Longshore Current 115 4.5 Relation of Normal Force to Moving Force ...... 123 4.5 Plot of Sand Distr'bution Across the Surf Zone 135 4.7 Sediment Suspension Due to Cb) Long Periods, High Fall Velocities, (c) Short Periods, Low Fall Velocities li+O LIST OF FIGURES ( Continued ) Figure Page 4.8 Vertical Forces on a Grain Particle About to be Lifted from the Bed 144 4.9 Relationship Between Sediment Mixing Coefficient and Mixing Parameter — g — ~ 150 4.10 Relationship Between Sediment Concentra- tion and Lift to Weight Ratio Parameter 151 4.11 Postulated Values of Efficiency Factor e, 154 a 4.12 Variation of Dynamic Friction Angle with Solids Reynold's Number 155 4.13 Sand Transport Model Relationship 158 4.14 Best Fit Sand Transport Model for Fairchild Data 159 4.15 Best Fit Sand Transport Model for Field Data 160 5.1 Definition of Azimuth Angle Normal to Shore 9 , and Azimuth Angle of Wave Propagation 9 168 5.2 Relationship Between Direction of Wave Propagation and Direction of Longshore Sand Transport 171 5.3 Location of SSMO Data Squares Adjacent to the Florida Peninsula 173 5.4 Azimuth of Normal to Shoreline at Ponte Vedra Beach, Florida 175 5.5 Determination of Total Positive and Total Negative Littoral Sand Transport 177 5.6 Ideal Case of an Unstable Null Point 191 5.7 Ideal Case of a Stable Null Point 193 LIST OF FIGURES (Continued) FigTjre . Page 5.8 Stable Type Littoral Drift Rose Due to Unimodal Wave Climate 195 5.9 Unstable Type Littoral Drift Rose Due to Bimodal Type Wave Climate 196 5.10 Instability Formed Capes in Santa Rosa Sound 197 5.11 Comparison of Computed and Observed Wave Heights at Daytona Beach, Florida 201 5.12 Comparison of Computed and Observed Wave Periods at Dajrtona Beach, Florida 202 5.13 Comparison of Computed and Observed Wave Heights at Lake Worth-Palm Beach, Florida 203 5.14 Comparison of Computed and Observed Wave Periods at Lake Worth-Palm Beach, Florida 2Q^■ 5.15 Comparison of Computed and Observed Wave Heights at Naples, Florida 205 5.16 Comparison of Computed and Observed Wave Periods at Naples, Florida 205 6.1 Logspiral Curve Fit to Shoreline at Riomar, Florida 211« 6.2 Logspiral Curve Fit to Shoreline at Lacosta Island 212 6.3 Logspiral Curve Fit to Panhandle Coast . 213 6.1+ Orientation of a Shoreline Segment Having Equal Offshore Wave Energy 217 6.5 Wave Energy Rose for Ocean Data Square #12 off East Coast of Florida 219 6.6 Energy Diagram of Coastline with Equilib- rium Coast Shape 221! LIST OF FIGURES (Continued ) Fig^e Page 6.7 Solution Curves to Equilibrium Shoreline Equation for the East Coast of Florida 222 5.8 Equilibrium Shoreline Shape for Riomar, Florida 22l| 6.9 Solution Curves to Equilibrium Shoreline for Gulf Coast of Florida 226 5.10 Equilibrium Shoreline Shape for Lacosta Island, Florida 22? 5.11 Equilibrium Shoreline Shape for Panhandle Coast of Florida 228 6.12 Equilibrium Shoreline Orientation for Shoreline Protected by Offshore Break- waters on the East Coast of Florida 230 6.13 Definition of Sign Convention for Shore- line Modeling 233 6.14 Solution Curve for Case of Sand Buildup at a Structure 239 6.15 Solution Curve for Sand Buildup at Struc- ture (After Time Sand has Reached End of Structure ) 243} 6.16 Beach Nourishment Plan Shapes for Beach Fill 242 5.17 Beach Nourishment Plan Slopes for Triangu- lar Initial Distribution 2m 6.18 Beach Noiarishment Plan Shapes for Gap in Beach Fill , 24S 6.19 Solution Graph for One End of Semi-infinite Beach Nourishment Project 245 7.1 Tidal Pr ism— Outer Bar Storage Relationship for Highly Exposed Coasts 25S xii. Copyright 19 7 9 by Todd L. Walton Jr, LIST OF FIGURES (Continued) Figiire Page 7.2 Tidal Prism-Outer Bar Storage Relationship for Inlets on Sandy Coasts 25U 7.3 Tidal Prism-Outer Bar Storage Relationship for Moderately Exposed Coasts 253' 7.1+ Tidal Prism-Outer Bar Storage Relationship for Mildly Exposed Coasts 255 A.l Definitions for Conservairion of Energy Flux for Shoaling Wave 276 C.l Variation of Average Annual Total Littoral Drift with Beach Orientation — Fernandina Beach to St, John's River, Florida 295 C.2 Variation of Average Annual Total Littoral Drift with Beach Orientation — St. John's River to St. Augustine Inlet, Florida 296 C.3 Variation of Average Annual Total Littoral Drift with Beach Orientation — St. Augus- tine Inlet to Ponce de Leon Inlet, Florida 297 C.^■ Variation of Average Annual Total Littoral Drift with Beach Orientation — Ponce de Leon Inlet to Cape Kennedy, Florida 298 C.5 Variation of Average Annual Total Littoral Drift with Beach Orientation — Cape Kennedy to Sebastian Inlet, Florida 299 C.6 Variation of Average Annual Total Littoral Drift with Beach Orientation — Sebastian Inlet to Fort Pierce Inlet, Florida 300 C.7 Variation of Average Annual Total Littoral Drift with Beach Orientation — Fort Pierce Inlet to St, Lucie Inlet, Florida 301 C.8 Variation of Average Annual Total Littoral Drift with Beach Orientation — St. Lucie Inlet to Jupiter Inlet, Florida 302 xxii LIST OF FIGURES (Continued) Figure Pag^ C.9 Variation of Average Annual Total Littoral Drift with Beach Orientation — Jupiter Inlet to Lake Worth Inlet, Florida 303 C.IO Variation of Average Annual Total Littoral Drift with Beach Orientation — Lake Worth Inlet to Hillsboro Inlet, Florida 304 C.ll Variation of Average Annual Total Littoral Drift with Beach Orientation — ^JiilJ_sboro Inlet to Cape Florida, Florida 305 C.12 Variation of Average Annual Total Littoral Drift with Beach Orientation — Perdido Pass to Pensacola Bay Entrance, Florida 306 C.13 Variation of Average Annual Total Littoral Drift with Beach Orientation — Pensacola Bay Entrance to Choctawhatchee Bay Entrance, Florida 307 C.14 Variation of Average Annual Total Littoral Drift with Beach Orientation — Chocta- whatchee Bay Entrance to St. Andrew Bay Entrance, Florida 308 C.15 Variation of Average Annual Total Littoral Drift with Beach Orientation — St. Andrew Bay Entrance to St. Joseph Bay Entrance, Florida 309 C.16 Variation of Average Annual Total Littoral Drift with Beach Orientation — St. Joseph Bay Entrance to Cape San Bias, Florida 310 C.17 Variation of Average Annual Total Littoral Drift with Beach Orientation — Cape San Bias to Cape St. George, Florida 311 C. 18 Variation of Average Annual Total Littoral Drift with Beach Orientation — Anclote Keys to Clearwater Pass, Florida 312 C.19 Variation of Average Annual Total Littoral Drive with Beach Orientation — Clearwater Pass to Tampa Bay Entrance, Florida 313 xiv LIST OF FIGURES (Continued) ^ig^e Page C.20 Variation of Average Annual Total Littoral Drift with Beach Orientation — Tampa Bay Entrance to Big Sarasota Pass, Florida 314 C,21 Variation of Average Annual Total Littoral Drift with Beach Orientation- -Big Sara- sota Pass to Venice Inlet, Florida 315 C.22 Variation of Average Annual Total Littoral Drift with Beach Orientation — Venice Inlet to Boca Grande Inlet, Florida 315 C.23(a) Variation of Average Annual Total Littoral Drift with Beach Orientation — Boca Grande Inlet to San Carlos Bay, Florida 317 C. 23(b) Variation of Average Annual Total Littoral Drift with Beach Orientation — Boca Grand Inlet to San Carlos Bay, Florida 318 C.21+ Variation of Average Annual Total Littoral Drift with Beach Orientation — San Carlos Bay to Wiggins Pass, Florida 319 C.25 Variation of Average Annual Total Littoral Drift with Beach Orientation — Wiggins Pass to Cape Romano, Florida 320 C. 26 Modification of Wave Data for Waves Paral- lel to Coastline 293 LIST OF SYMBOLS A = mixing parameter constant = inlet cross sectional area at throat of tidal inlet a = wave amplitude B = proportionality factor associated with bed load transport Bg = proportionality factor associated with bed load transport inside the surf zone B^ = constants of longshore velocity distribution ^i' ^i ~ functions of bed load portion of sand transport equation. c = wave celerity c^ = wave celerity at wave breaking Cg = speed of wave energy propagation c^ = wave celerity in deep water C = concentration in dry weight of sediment per unit weight fluid = drag coefficient = friction coefficient for flat plates C-j^ = life coefficient = Chezy friction coefficient d = mean sand grain size dgp = median sand grain size D = n + H, total depth of water XV i = total depth of water at location of breaking waves = depth of water at limits of longshore sand movement e^ = efficiency coefficient associated with bed load transport e^ = efficiency coefficient associated with suspended load transport E = energy density E* = quantity proportional to energy flux density E^ = energy density in deep water f = friction factor f = frequency of occurrence f^ = friction factor defined by energy dissipation for waves f^ = friction factor defined for oscillatory flow = functions of sand transport equation F = energy density flux F . = horizontal force per unit area due to slope of '^'^ free water surface g = acceleration due to gravity G = gravity force on a spherical sediment particle h = depth below still water hj^ = depth below still water level at wave breaking h^^ = height of beach berm above still water level h^ = depth below still water level in deep water h^ = total depth of active beach profile (in longshore sand transport) H = wave height H, = wave height at wave breaking xvii ■ wave height in deep water ■ significant wave height index corresponding . to horizontal coordinate in the x-di^^^ection inmersed weight sediment transport rate per unit width immersed weight sediment transport rate per unit width (bed load) immersed weight sediment transport rate per unit width (suspended load) total immersed weight sediment transport rate for surf zone total immersed weight sediment transport rate for surf zone (bed load) total immersed weight sediment transport rate for surf zone (suspended load) index corresponding to horizontal coordinate in the y-direction wave number ratio between, set-up slope and beach slope constants friction coefficient refraction coefficient shoaling coefficient rip current spacing wave length lift force on a spherical sediment particle wave length in deep water mass of bed load sediments X viii m = mass of suspended sediments M = summation constant = + M^, total mass transport per unit width or total mean momentum per unit area = mass transport per unit width associated with fluctuating motion n = transmission coefficient N = dimensionless lateral mixing coefficient N* = normal directed, component of energy density at shoreline p = pressure p = dimensionless lateral mixing coefficient p^ = porosity = power/unit area available for bed sediment transport = "longshore" energy flux Pg = power/unit area available for suspended sediment transport P* = parallel directed component of energy density at shoreline Qi^ = total volumetric longshore sand transport rate r = roughness parameter r^ = ripple parameter = total average resistance force R = ripple height R = equivalent of Reynolds Number in solids mixture s s = fraction of incoming energy flux available to the longshore current xix proportionality factor associated with suspended load transport specific gravity of sediment proportionality factor associated with suspended load transport inside the surf zone excess momentum flux tensor (radiation stress tensor) function of suspended load portion of sand trans- port equation time wave period gradient of "radiation" stress due to periodic wave motion water particle velocity maximum due to wave motion at the bed total water particle velocity fluctuating water particle velocity component water particle velocity due to wave motion mean velocity, component normal to the beach mean velocity of bed load transport in the long- shore direction mean velocity of suspended load transport in the longshore direction + , total mean transport velocity mean transport velocity associated with mean motion mean transport velocity associated with fluctu- ating motion resultant velocity vector of combined wave and current motion velocity component parallel to the beach xx: dimensionless velocity parallel to the beach velocity component parallel to the beach at loca- tion of breaking waves mean velocity (across surf zone) parallel to beac: vertical velocity component fall velocity of sand grain horizontal coordinate perpendicular to the beach dimensionless distance from the shoreline width of the surf zone horizontal coordinate parallel to the beach vertical coordinate incident wave angle incident wave angle at breaking incident wave angle at wave generator incident wave angle in deep water bottom slope specific weight of fluid dimensionless suspension height kronecker delta energy dissipation function energy dissipation due to bottom friction vertical sediment mixing coefficient surf similarity parameter water surface elevation mean water surface elevation azumth angle of wave ray XX i ®n ~ outward directed normal of shoreline ic = ratio between breaking wave height and the depth of water at breaking IC" = shoreline diffusivity value X = phase function A = linear spacial concentration in sediment-fluid mixture |i = dynamic viscosity of fluid ^ei ~ dynamic -eddy viscosity of fluid V = kinematic viscosity of fluid C = roughness parameter = water particle excursion due to wave motion at bed P = density of fluid Pg = density of sediments a = local wave frequency ^ij ~ shear stress '^hi " ^^^^^ stress at the bed (j) = dynamic friction angle *^'s ~ static friction angle = angle of repose for sediment $ = wave sheltering angle X = dimensionless coefficient dependent on wave and sediment parameters Y = velocity potential u = wave frequency = resultant angle of energy density at sheltered shoreline point xxii ub scripts b = breaker line or bed load h = bottom m = maximum (time wise) o = deep water s = suspended sediment w = wave xxiii Abstract of Dissertation.-Er>e^ exited to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy LITTORAL SAND TRANSPORT ON BEACHES By Todd L. Walton, Jr. June 19 79 Chairman: Dr. B-. A. Chris tensen Co-Chairman: Dr. T. Y. Chiu Major Department: Civil Engineering The transport of sand along beaches within the surf zone is considered in this dissertation. The driving forces for sand movement are considered to be the breaking water waves which transfer a part of their energy to the shoreline made up of noncohesive sandy material. The primary goal of this dissertation is to develop a model of sand transport using the excess momentum flux of a water waves approach as a driving force for the water cur- rents and an energistics approach to develop the sand trans- port created by the water currents. Both bed load and suspended load are considered in the model which is an outgrowth of the past work of M. S. Longuet-Higgins on "radiation" stresses within the surf zone and of R. A. Bagnold whose energistics approach to riverine sand transport is adhered to in the development of a similar model for littoral sand transport within the surf zone of sandy beaches. The equation for littoral sand transport developed in this dissertation has- a linear (in breaking wave angle) xxiv component' of suspeTid^d -s-ediment -tiraTisport , and a nonlinear (in breaking wave angle) component of bed load sediment transport. By using published experimental data, two unknown coefficients for the model are found. Existing data used in model correlation indicate that the model is reasonably valid. As a secondary goal of this dissertation, the model is used to develop sand t^^ansport values along the sandy beach portions of Florida. The computed values of sand transport are compared to existing estimates with reasonable results. Additionally, a new method of presentation of littoral sand transport computation results is suggested which pro- vides a means of evaluating stability/ instability features of shorelines. An example is given of how bimodel wave climates of elongated bays may lead to the numerous insta- bility features (giant cusps) seen in these bays. Additional analytical modeling is proposed using linear formulation (in wave angle) of the sand transport equation and is found to be useful in predicting the shoreline shapes of many coastal features. XXV CHAPTER I INTRODUCTION 1 . 1 Introductory Note Sandy Beaches have long enjoyed great popularity with both the residents of coastal areas and the enormous influx of tourists se-eking out these vacation meccas . It has long been recognized that these same beaches, an increasing source of income for coastal zone citizens, are in serious trouble due to erosion. Preservation of these beaches is not only desirable aesthetically, but is also an economic necessity. The economic significance of the erosion problem can be seen (at least in one state) from costs shared between the State of Florida and the Federal Government in order to preserve Florida beaches. As of 1970, estimated first costs of authorized Federal beach improvment projects in Florida amounted to over 76 million dollars for 108 miles of beach. Estimated first costs to correct all the existing erosion problems in Florida (includes authorized and unauthorized projects) amounted to over 113 million dollars for 2 09 miles of ocean shoreline (1). Of approximately 1000 miles of sandy beaches in Florida (see Figure 1.1), the annual quantity of erosion in the 1 2 GEOR Gl A C Ferrcndina . ; M^^st. Marys River St. Aview's Inlet JockscvTville ~' John's River St. Ajgustine Inlet Matanzas Inlet Daytona pQuca Da Lecn Inist Cops Curm-eral LEGEND ANMJAL SHCRajNE RFCE3SCN ~~ 1 20 Ft. Plus iO-20R. 5-)0Ft SH8 2-5 Ff. « I-3F1 I Q-l Ft. C5 Q Sedcsfian Inlet Ff. Fieme Inlet St. Luds Iniei Jupitar Island J'jpifBr Inlst rbi.71 53ccij Inist ■3cynk>n iniet 3cca .Raicn Inlet P^. E'.'iinjkzlss In (at BalKSrs Hciiiovfr Inlet Micmi 0 Scale 20 50 Milfl .J i_ Key Vtet^ 0 20 60 KiiciT^tars Figure 1.1. Erosion Situation in Florida 3 nearshore area has been estimated at 15,000,000 cu. yds. per year, with over 20% of the beach shoreline in a critical state of erosion. Factors that influence beach erosion are LITTORAL SAND TRANSPORT ON BEACHES By TODD L. WALTON JR. A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 1+ 1910 !92D 1930 1940 1950 I960 1970 TIME (Yecrs) Figure 1.2. Relative Rise in Sea Level In Florida (from Hicks ' (.2)) 5 Dunea Beach Sea Level Aft a* Rise Inttid .Sear Level Initial Bottom Profile [ Hypothetical Bottom Profile After Rise in Seo Lave! Actual Bottom Profile After Rise Of Sea Lav^ And The C^jcaTtftafivft Balance Bet»*eeR Erosion And Bottom Deposits After Sea Level Rlsa Limiting Depth Between Predominant Neorsftoref And Offshore Material Littoral Drift Characteristics Figure 1.3. Bruun's Concept of Beacli Profile Response to Sea Level Rise (from Bruun, P. (3)) 6 volume of sand from the. .upper, heaxih profile will slough off to maintain an equilibrium bottom profile offshore. This hypothesis was first discussed by Bruun (3) who developed a quantitative relationship for the rate of shoreline retreat in terms of the rate of sea level rise which, for Florida, amounts to about 1 to 3 feet of beach recession per year. As many areas of the coast appear to be relatively stable (i.e. much less than 1 foot -erosion per year in the short term), it is reasonable to postulate that either the hypoth- esis is wrong or that a trend of shoreline erosion in response to rising sea level is not gradual but rather takes place during more severe wave activity such as occurs during hurricanes or extra-tropical storms. In areas of normally low wave energy, rapid response to high wave events does occur and lends credulity to the above theory. Severe erosion occurs on our shorelines during storms and complete recovery of the lost sand is never made to the beaches. An example of what can happen during a hurricane is shown in Figure l,i+ from the work of Hayes (U). This figure docu- ments the potential for a large storm (Hurricane Carla) to transport tremendous quantities of material offshore out of the zone of all but extreme wave activity where it will not be returned to the beaches. In this storm, over 100 feet of dune system was virtually destroyed and a sand layer varying in thickness from 1 to 9 cm was deposited out to depths of 7 20 Ft. Post-Caria Gi Bed Thickness l->9cm I— S-9cm I— 3- 6cm 1- I -3cm- m - Sand Layer (Post-Carla) 2--^-Ma\or Hurricane ■~" Channel GULF OF MEXICO Artifical Channel — Hypothetical Density Current Flow Pattern Contour Interval 0 5 20 R. Figure 1.4. Sediments Deposited off Central Padre Island During and After Hurricane Carla on a Mud Bottom (from Hayes (4)) 8 120 feet by a combination of hurricane waves and currents. Figure 1.5 presents the results of a more recent study of the beach erosion occurring during Hurricane Eloise which made landfall just west of Panama City, Florida in September 1975 (5). The dune erosion profile shown is the result of a composite profile taken over a 20-mile stretch of beach from Panama City Beach to Destin, Florida. Over 7 5 feet of large 15 to 20-foot primary dune system was virtually destroyed by this storm. It will take years before part of the dune system will be rebuilt naturally from the sand deposited offshore during this storm, and from experience gained from other hurricanes the beach-dune system will never recover completely. Sea level rise coupled with severe wave events is not the only reason for our erosion problem though. Another major contribution to erosion is our inlet systems and their corresponding navigation channels either natural or artifi- cially cut through the littoral zone. In the sandy beach littoral zone of Florida alone there are 5 7 inlets. Fourteen of these inlets have Federally authorized navigation projects with authorized channel depths extending to the ocean or gulf of over 20 feet. There are at least 14- more navigation projects with authorized channel depths of 10 feet or more. For comparative purposes, natural controlling depths on the outer bars of "unimproved" inlets are on the order of 6 to 8 9 £=: 0 d) 3 +J O) ^ — ~ CQ tn ■ 1- (O o s- r— 3 LU Q- S- cD :r c: Q. o -a •1- E i- « i_ 3 3 O) 4-' £ 1- O <c s- -c — - c (13 ra T3 CD -i- S_ S- o o ■ M- ■— <D U. CD n •1- +J to O <D CL. -a 3 u ca (D fO CO s- c (1) cs N0I1VA313. 10 feet. Many of our Feder^al. navigafion projects are not natural channels but have been cut directly through the corresponding barrier islands such as St, Lucie Inlet on the lower east coast of Florida. The effects of this inlet will be mentioned later. When a channel is either cut through a barrier island or dredged below the natural existing depths, the flow of water through the channel to the bay (or lagoon) on flood tide and to the oceanCor gulf) on ebb tide is increased leading to an increased capability of the channel to flush sand to its inner bay system or outer shoal system. The channel also acts as a barrier to littoral sand moving along the coast which eventually works its way into the channel and then migrates to either the inner or outer shoals of the inlet. The sand in the interior of the bay systems cannot work its way back out as there is no wave activity to agitate the- sediments into suspension such that the water can carry it out. Thus, the bay shoals of these inlet systems act as net sinks to the beach sand system. Volumes of material residing in the inner and outer shoal systems have been shown to be substantial (6). An example of the amount of sand stored in Florida's outer shoal systems is shown in Table 1.1. In Florida, considering present erosion rates, it has been estimated that over 200 years worth of sand resides in the outer shoals of these inlets (6). The sand moves to these inlets by means of littoral 11 O >, U > CO U >H en H i •ri H u fn O 0) fC to > ■H m m u u rt) dJ Q} >i ■P +J tU (T3 O H 0 o •H X) O CO c H M O Ct3 4h O O w rH M T3 n3 0) T3 rH •H m rH s ^ CO M •H to CO fH 0) 03 C3 H c ra Oh 1 e CO < M 1 m a C M <U (U •H W 3: ■H o rH n} O U "-^ eq J O >! > >i 00 m rH o; fO w +-> FQ X! fH U (T3 (U > 3 o > IT) 13 S ■H H it3 (tJ r-l O O o o o •rH 0) OCNCSICNJ-J-^;:f^aDlOlDa3rH LnuOLrjLOLOLnmLOLOLOi-ni-nLniD HiHHf-i'HHrHHrHrHrHrHrHH o •CP H OOOOOOOOOOOOOO --IrHrHrHHHrHrHr-IHrH'-HrHrH LD ID O to J- CO cn o- CD <n H H rH UD ID CD <£) O O O O rH rH rH rH M to o a 14h H 3 O O O CO o CO T3 ■H to (0 CO O to /3 Oh bO C S O (P J 2; X X X X X X X X X X X CO CO CD O o CD O) :3- CD CO o CO CO CX) CD o CO CN cn 00 ij- o 00 o O CO en CN en O O i-H en H CS rH CO CO CO (0 M CO CO CO rO to CO CO tO (U tO > CG to to +J CO CO eu o CO to +-> to 0) 0) to CO >-H CO CO Oi o a, rH (0 10 to to — ' lO to CO C H c 0- o CL, to o lO +-I 1— 1 H to rH CO a .c •rH to CO CO to bO OJ O > to to to c to CO •H o lO •H •rH CJ Oh •rH 0 c •rH CX, to •H >, bo CO CJ a- X) bO tu bO u bO •H to O to •H •rH 0 •H pa > PP o Pi P3 2; 3: t3) m ac3 to O in J- CD c- CO O en IT) un LO CN CO :t cn CD CD CD CD CD CD CD r-H rH rH rH H rH rH 1 1 CO 1 CO J- CO Ln Ln cn LT) CD CD CD CD rH H rH r-H rH rH J- IJI CD CD CD CO rH CO CO J- d- CO CD CD CD CD rH H H H I or H CD UO CD CD o O O O O O o o o O O o O rH H rH rH rH rH rH >H r-{ rH rH H X X X X X X X X X X X X X r- CD o o o CD o CN CN o O cn CM CD CD CD CO o CD CD 00 O CD O O O J- rH CN H ID CO lO CD O tD IT) H > •H Pi - (U CO >i to fn O to +J • CO -H rO CO W > •H Td oi c 3 O o 00 tn o 0) o c to !^ c ■p -p 0) CD rH rH C C M M tU c c o •H a) -P CO 3 0) bO Td 3 < OJ o . c -P o CO 0-c > o rH 3 to 0) CO HJ U •H 0) 3 to H-- O CO to ^^ O to o n: to rH rH 3 O C3 O to CO D. 0) CU P-. 3 to CO to •p to u ^ c rH >, C to 3 m i-o to CJ P4 o t3 C to rH CO bO O Q CO ' CO rO to CO 0-1 CO CO to lO C PU 0-1 10 ■H -P P TO CO to C <U lO M 3: W 12 o o O o o o o o ncl o o o o o o o o o o o o o o o o tn o o a o o o o o CD CO o CO t>- CD IT- CO o o O to rH oo CO tH 0 cn CO CO CO ■rH cu cu cu (U 0) (D H (D iH rH iH H rH rH (U H •H •H •H •H •H •H e S s e s O -H CO CO CM CO o 00 CM H o o o o O o o o (U H o o o o o o o o H '-^ o o o o o o o o •HCO IH t3 CN o o o o o CO o o CO CO LO un o CD o > M_l w CD CO r-- CO CO o 0 *^ #\ CM CM oo r-i cn <T> rH ■P CD CD CD r- i>- CD CO LO C cn en cn cn CD CT) CD O H CU H rH iH iH H H rH rH tu rH 6 •H (U !h >, >, >i Ph o •H O (U H rH 0) rtJ ^ S rH Cm s < S cu CJ > <13 O Q o m Q) •n Fl •H O ft ft ne ■H CO p >, CO G 03 •H CU 0 CO S CO OJ CO o 0 03 •H rH •H CO rH PQ Ph ■H 2: ti-. >, 3 0) OJ >, O in rH >4 G P 0 G 03 m G fl3 rd >, H iH OJ H <u CO 0 O CQ Is av M 0 c 03 <U G o •H 0 ■H -H G C^l P rH •H ■H CO •H (13 0 P W) 03 !^ O G ft f-i 0) U O d 03 •H 03 rJ 0 > E-i a:; 13 sand transport along the beaches ; the primary driving force being wave activity. Figure l.S shows an example of inner shoal sand trapping for St. Lucie Inlet which was cut in 1892 through the barrier island. As noted by the solid line in Figure 1.6, the total sand deposited in the inlet over the years 1892 to 1930 amounts to over 9 million cubic yards of sand. It appears £rom the trexid shown in this figure that inlets shoal rapidly in their early years and eventually reach an equilibrium inner shoal area (7). The trend for outer bar shoals of inlets is not so apparent. Figure 1.7 shows a relationship proposed by Walton (8, 9) between the volume of sand stored in the outer shoal of an inlet and the size of the inlet as determined by its cross sectional throat area (which is related in turn to the volume of water flowing through the inlet). Three curves are given express- ing various severity of wave activity at the inlet. The following sand storage equations postulated by regression analysis by Walton (8, 9) are given for the three wave conditions : 1 28 V = 33.1 * heavily exposed inlets 1 2 8 ¥ = 40.7 A^ * moderately exposed inlets 1 2 8 V = 45.7 A^ ■ mildly exposed inlets where ¥ = volume of sand stored in the outer bar in cubic yards, and A^ = cross sectional area of the inlet in feet squared . 14 cn g cn o CL hi o < cn (0 a cn GO 00 ^ 2 220- 200 180 160 140 120 100 80 60 40 20 cn CD "S — a cj cri <s cn a CD CO CD pTrend Line for Shoaling Pattern in St. Lucie Inlet. ( From 2 miles North of In let to I mile South of niet in I.C.W.W.) I — Trend Line for Shoaling Pattern n St Lucie Inlet. (Vicinity of 3/4 mile Radius of Inlet.) 1883 18s: 1900 1910 1920 1930 1940 1950 1960 YEAR Figure 1.5. Deposition of Sand in the Interior of St. Lucie Inlet (from Dean and Walton (7)) 15 X . — r— 03 • O) U- U- • . r— . (/) fa Lu X - - - trt — • • H— brt ui vl (/I ^ f-™ OJ • - m (O tn CC'rtaO- Q- .— 1/1- C3 (/» * -C fO O > O U *TJ OJ O tC i/lt/»i— J3 (Oi/lC •TJ > C >,£rrt3*J CT»Q_C ' LU 1/1 U -a c -o c; > 3 «e -a u "O oj oi-c £ c o ro m O fT3 2 O u > W (/I o o a: j3 u c _2 >> tj u. 5 a o t- o o < =3 d n3 <: OJ U1 a o > :e u I ■in s o o "CVJ — o — tti a o cn 8 -o c a (J z LL. u. U. ■ -3 O . lO • -3 x: 3; u. c u . c •o ■ u. o c OJ 01 1 — - cn • > O — ^ 1 — LLi cn a) ^ cx « - i. o UJ o "c O U OJ c Ol c: a. c ^ ii <: -5 (3 ^ 13 O -4-> at O) cn 3 w cn u c CI -4-> O — (/I iTJ C 0) OJ c (/> "3 I. a. ■t-i rt5 0) 0) — S- o (H i. ta <z ra 3 a 1 o ts a. m in yD r-». o ai a — OJ "B c _ o i O O i 4 i i O O ifJ <M (>4 ~ o poo o o lo o crt O c: CO rO OJ c cn o S_ f— O (t3 on E s- o « i. cn u. s_ O) CO ■M 4-3 o ■— <u 3 O 4-J O) n3 s- 1/1 cC Q. I — c o O -r- •r- -M -P O (O O) 1— (/5 OJ 0) 3 16 As seen -in the . aLove. equations j the more severe the wave activity on the outer bar, the smaller its storage capacity is — i.e. wave activity limits the size of the outer shoal area by driving shoal material back to the beaches. The size of the inlet is the main controlling factor though as larger inlets store more sand. Thus deepening an inlet and consequent enlargement of an inlet's tidal prism may well cause additional shoaling on the outer bar if bar storage equilibrium has not been reached. The sand necessary to make up this additional storage volume must come from adjacent beaches. Figure 1.1 is a view of the erosion situation along Florida's coast in 196 3, but is similar to the erosion situation today. The large black spikes where the critical shoreline recession exists are at locations of inlets thus confirming our expectations that inlets cause erosion to adjacent beaches. Of course, it is also well known that in areas of a predominant net sand transport along the beaches, improvements such as jetties at inlets can cut off the natural flow of sand and thereby starve downdrift beaches of sand. In Florida, on practically every south side of an inlet (downdrift side), erosion is excessively high. It is not uncommon in Florida to have large stretches of shoreline adjacent to inlets undergoing recession at rates upwards of 10 feet per year. 17 Another cause of "apparent" erosion to our shorelines is that of barrier overwash. Many of our low barrier islands are very susceptible to wave action occurring over the barrier island during periods of high tides with a consequent driving of sand into the bay systems as overwash fans. Very little is known as to the quantities of sand .lost during overwash events (10). / Causes of erosion are then due to the two modes of sand loss; offshore sand lossybrought about by the onshore- offshore sand movement mode,^and longshore sand loss (which moves sand toward inlet traps) brought about by the "littoral" sand transport occurring due to wave activity on sandy beaches. Erosion due to both modes of motion can be de- scribed by a simple continuity of sand equation and applied to predict shoreline changes. The topic of this thesis though will only deal with the longshore sand transport, "littoral" drift, which appears as the major cause of erosion when man interferes with the beach -_( 11-2 3 ) . 1 . 3 Ob j ectives This thesis will specifically deal with a predictive model for determining sand transport along sandy beaches as caused by water wave activity. Although in the vicinity of inlets or estuaries other factors such as tidal currents come into play, only the natural forcing function of gravity water waves will be dealt with in this thesis. The model 18 will be applied to the jnovement .O-f. cohesionless material (sand). Using the predictive model, sand transport will calculated for Florida's shoreline and a linearized sand transport model will be considered for use in predictive techniques for shoreline changes. CHAPTER II REVIEW OF SURF ZONE MODELS 2 . 1 Review of Existing Longshore Sand Transport Models Numerous longshore sand transport models exist in the literature although none give better than an "order of magnitude" agi^eement with exisiring field and laboratory data. As the present models are mostly based on wave param- eters as the driving forces, it is important to note that no sand transport would occur in most of the models if wave action were not present to both stir up the bottom sediment and drive the longshore current. Most of the more popular surf zone littoral sand transport models (24-38) are listed in Tables 2 . 1 and 2.2. Equation 2.1(a) was historically arrived at by attempts to relate the volume of sand transported along the coast to a wave power term referred to as the "longshore energy flux, P^. Early researchers made empirical formulations of the type: = Constant • pj (2.1) Caldwell (24) first found an expression based on data with an exponent n - 0.8. Other researchers using other data made separate attempts to refine this equation with various values of the constant and the exponent, n. Many of these 19 20 Table 2.1. Littoral Sand Transport Models. Reference (2.1a) = K^^ (25) or (2.1L) = (26) where = Q^Y(Sg - 1) (l-p^) H J. o c , COS a, „ (2.2a) I = K, \ ^ ^ (27) s 2a w f s or 2 YH C , cos CL _ (2.2b) I = ^ V (27) s 2b w ^ 4-/3 cos a, n /o tang sin a, (2.3) Q, = K3^ — ^ -^^ ^ (28) d (2.5) S=i, ti =— ±— .(B (^)^^.S -1)^ ^ ^ ^ gd-fi- ) ^ "hwm ^ V ^ -l)d (2.6) = B*d ^ g= . exp -(-A* § ) (31) r^(— ) (1 + — — ) ) z 21 Table 2.1. Definitions. Kij = constant = volumetric rate of sand transport (1^/t) Pj2_ = longshore energy flux (~) - immersed weight sand transport rate A) t 3 - beach slope angle = wave breaking angle w^ = fall velocity (1/t) = wave breaking height (1) Y = specific weight of fluid (m/£^t^) T = wave period (t) d = grain size (1) ^^b' ^gb ^ "^"^^ speed at breaking (1/t), wave group spe at breaking ^2 = Chezy friction factor = ripple parameter = shear of flat sand bed total shear A* = empirical coefficient ( = 5 BijkerOl}) = empirical coefficient ( = 0.27 Bijker CSU'^) C = roughness parameter ( = 0,045 C BiikerOl)) P = fluid density (^/l^) Pg = sediment density (^/l^) g = acceleration of gravity Cl/t^) ^hwm ~ ^^^i^um bottom orbital velocity (1/t) Pq = porosity of sediment in place S = specific gravity of sediment 22 Table 2.1. Definitions (continued) K - constant related to ratio of wave breaking height to depth E = energy density of waves (5^) t^ 1^ - xmmersed wexght transport rate per unit width (— ) t^' ^s' " efficiency constants 1, t, m = length, time, mass 23 0) o c u 0) Pi ta C 3 C CO CO ID CN CO CO CO CO CO CO CN a -H (fl H ■P rH td d) H -p nj -P <u (U 0) Q) to tiO n3 rd IXJ 6 g o iH > •H •H 03 •n u ■1 — 1 !0 w CJ CO f— 1 l-H CO QJ ear >i ^ -M Mh u >> >i C ■H iTJ o O nJ IT) O 0) <D (I) CD "O \ \ 6 \ ^ >, w CO =& 4h \ \ \ =rt: ■P & 03 (U >. Mh >i >, -H n3 n3 O Q) m IT) \ s >) -\ >> o CO CO CO CO CO \ CO H e e 6 >1 03 O m CO o o o O H o O r-H H to H CO H O o o o o CN 03 CN CD CN UD CD CO o rH O TO O o o O o O o c <u Q) Lp OJ Pi C^ CO CO CO CO CN ^ ' T) C to o w c IT! n) to IT) o > 6 OS e n) C w 0 w 00 M a o o CN CN d- 00 o O o o o a, to II Ok ti CO CO r-l a a •H «i a m e ■H -d I H 24 formulas are sununarxzed iji . TaiJLe. .2 ^ 2 . Later CERC (Coastal Engineering Research Center) revised the formula with an exponent n = 1.0 which is now in use as the standard littoral sand transport equation used by the Corps of Engineers (25). Limitations in the model exist though due to its neglect of factors pertaining to the material transported such as grain size of sediment, density of sediment, porosity of sediment in place- and ■ -temperature effects which have been shown important in suspended load in rivers and can make a factor of two difference in the transported load where a wide variation in temperature exists (39). Another short- coming of this formula is that beach slope is neglected. The constant in equation 2.1(a) is dimensional. A modification of equation 2.1(a) was first made by Inman and Bagnold (M-0) changing the volumetric rate of sediment transport to an immersed weight sediment transport rate as in equation 2.1(b). This allows for a dimensionless constant (more physically appealing) as well as accounting for grain density and sediment material porosity effects. A plot of this equation with considerable laboratory and field data is shown in Figure 2 of Inman and Frautchy (25). The dimensionless coefficient K^^ of equation 2.1(b) was found (26) to be K-, , = 0.25. Other researchers have lb given various values of K.^^^ equal to 0.2 (32); 0.4 (37); 0.77 (38); and 0.8 (25). Many of these constants are summa- rized in Table 2.1. 25 Longuet-Higgins (^.1) has . criticized the above formula- tion as the parameter called "longshore energy flux" is neither the longshore directed component of energy flux nor the onshore component of energy flux. As the model does not differentiate between bed load type of transport and sus- pended load transport, both modes of sediment motion are included in this model. Equation 2.2 was postula-ted in a heuristic model by Dean (27) on the assumption that sane transport moves as suspended load. This formula does take into account many of the missing factors of equations 2.1(a) and 2.1(b), such as beach slope and temperature (via grain fall velocity), as well as grain density and porosity of sediment. Additionally the model can be used for the prediction of sand transport when a longshore current other than that driven by waves is present. An additional parameter of importance in this model is that of a bottom friction factor, C^, which is dependent on inviscid flow bottom orbital amplitude and velocity. This factor will be mentioned again later. The critical shortcoming of this model is its inability to predict bed load which has been noted by others (42, 38, 43) to be a major portion of the total sediment transport under conditions of low wave steepness and large sediment grain sizes 26 Equation 2.3 was postulated by Iwagaki and Sawaragi (28) on the basis that the average littoral transport is proportional to the shear velocity of the mean longshore current. They use a Kalinske-Brown type of sediment trans- port formula. The mean longshore current used in the formu- lation was that of Putnam, Munk , and Traylor (44). Limita- tions in this formula are evident in the lack of grain density considerations, sedinaent porosity, temperature effects (due to the use of grain size rather than fall velocity) and the dimensional form of the proportionality constant. Density and porosity considerations can be taken into consideration by a simple transformation used in equa- tion 2.1(b). Equation 2.U, postulated by Le Mehaute and Brebner (29) on physical reasoning and empirical information, has the same limitations as the previous equation along with the absence of two important parameters, beach slope and grain size. Equation 2.5 was postulated by Thornton (30) using an energistics approach as in Bagnold (45) for calculating the sand transport per unit width within the surf zone. Thorn- ton's model takes into account most all of the important sediment transport parameters except for sediment porosity but does not define the efficiency factors B or S which s s makes the equation unusable in a practical sense. Addition- ally, Thornton correlates the work rate of bed load transport 27 to the total available energy flux of the waves rather than to the energy expended in the longshore direction. As a result, the energy expended in moving sand back and forth (in the onshore-offshore direction) is included within the available power for longshore sand transport, while, in reality, no work is being done in the longshore direction by this energy .dissipation. Equation 2.6 has been po^stulated by Bijker (31) for bed load transport in the surf zone based on extensive laboratory tests and a sand transport model proposed by Frijilink and modified by Bijker (31). Weaknesses in the model are ex- pressed in the form of empirical coefficients which were determined on the basis of laboratory tests and give values ranging over two orders of magnitude. Additionally beach slope and material porosity are not taken into consideration. The lack of existence between an expressed form of bed roughness (in the surf zone) and the Chezy friction factor adds a qualitative factor to the model which makes it diffi- cult to apply in practical application. Bijker (31) proposes a suspended load as a function of the existing bed load. 2.2 Review of Longshore Current Velocity Mode ls As most of the longshore sand transport formulas take into consideration the longshore current either explicitly or implicitly, a review of the longshore current models is pertinent. Most of the longshore current formulas predict 28 only the mean of the longshore current velocity with the exception of the Longuet-Higgins model (46), and the Bowen model (47). There are four basic approaches to the development of predictive equations for currents in the surf zone: conser- vation of mass, conservation of momentum, conservation of energy, and empirical correlation. Most of the more familiar equations, are presented in- Table 2. .3. An additional long- shore current model by Thornton (30) has not been included in the table but can be found in Reference (30). Some of the equations have been expressed in slightly different forms than originally published in order that they all incorporate the same parameters; however, these changes deal only with the geometric relation for the breaking depth and breaking wave height. The changes do not alter the currents predicted by the equations- Symbols, utilized in the equa- tions are listed in the preface. Much of the following review of these equations must be credited to Thornton (30). The equations will be reviewed in order of increasing analyt- ical usefulness with empirical equations having the least utility due to their lack of physical justification or rational. Empirical correlations — Two types of empirical equations have been developed. The first type employs physical reason- ing to determine the form and grouping of the important parameters which are then correlated with experimental data. Brebner and Kamphuis used both the energy and momentum equations to obtain reasonable groupings by dimensional analysis of the important parameters. Linear regression was then used to find the best fit for the longshore velocity to a large number of data that they had measured in the laboratory. The second, type of analysis employs multiple regression techniques. Sonu et al. (49) used this method to weigh the various independent variables collected in their field studies. They found that the most important variable affect- ing the mean longshore current velocity was the angle of wave incidence, and the second most important, although much less, was the wind. These results are conflicting with those of a similar analysis reported by Harrison and Krumbein (50) who, using data collected at Virginia Beach, Virginia, found the most important variable to be the wave period which proved to be insignificant in Sonu's analysis. Sonu also performed a multiple quasi-nonlinear regression analysis which showed the most important variable affecting the mean longshore current velocity to be the wave height. In a later study, Harrison (51), using another set of data col- lected at Virginia Beach, found the incident wave angle to be the most important, followed by the wave period, height, and beach slope, respectively. Harrison notes that the use 30 of empirical equations is limited in application to "similar" situations; it is not possible to extrapolate to different type beaches than considered in the empirical formulation. Another problem with using empirical formulations is that they are devoid of physical basis and, as such, can give spurious correlation and conflicting results .( 52 ) . Continuity considerations--Chiu (53) and Inman and Bagnold (40) .derived similar ..expressions using the continuity approach. These formulations are based on the fact that the incident waves introduce a mass flux of water into the surf zone which is then manifested as a spatial gradient in the longshore current. Both developments consider a plane beach of infinite length, implying that mass is uniformly intro- duced into the surf zone along the beach. The current will grow (since mass is continually being supplied to the surf zone), and at intervals, it is necessary that there be outflow from the surf zone unless the current becomes un- bounded. It is postulated that this outflow occurs in the form of rip currents which are evenly spaced along the coast- The equations thus contain an unknown parameter — the spacing of the rip currents. Unfortunately, few measurements have been made of rip current spacings so that the use of these equations requires additional experimental data. It should be noted that, due to the mass flux of waves, there is always transport of fluid into the surf zone and 31 -that,, in all the physical models .whether considering a mass, energy, or momentum approach, the mass flux must be accounted for in order to obtain a bounded solution. Thus, the assump- tion that the mass transport is uniformly returned across the surf zone, is at least implied in all the developments which do not include concentrated return flow by rip currents. Chiu and Bruun (53) also considered the case where rip currents are absent and the return flow is distributed uniformly over the vertical plane containing the breaker line. He reasoned that waves breaking at an angle to the beach contribute mass to the surf zone and locally raise the mean water level as the breaking wave crest propagates down the beach. This results in a slope of the water surface between crests which creates a longshore current. The longshore current is balanced by bottom shear stress related to the velocity through the Chezy formula. Galvin and Eagleson (54) reasoning from the continuity approach, equated a hypothetical mass flux across the breaker line proportional to the mass contained in the longshore current. Using both field and laboratory data, the two mass fluxes were correlated. Energy considerations--Putnam at al. (44) also derived a mean longshore current equation from energy considerations alone. The derivation equates the changes in energy flux to the frictional energy losses parallel to the beach. A 32 difficulty, with the resulting equation, is that it involves two undetermined constants, a friction factor f and the percentage of the wave energy available to the longshore current, "s," which makes the equation very difficult to apply. Momentum considerations— Putnam, Munk , and Traylor used both the energy and momentum equations to derive the first rational equations JLor^^jdescribing longshore currents . They considered the flux of mass and momentum into a control volume of differential length bounded by the breaker line and the shore. The change in momentum flux across the breakers directed parallel to shore is balanced by the bottom shear stress. Solitary wave theory was used to calculate the momentum of the breaking waves . In this manner, they obtained an expression for the mean longshore velocity related to the angle of wave incidence , breaking b wave height H^^ , bottom slope tanB, wave period T, and fric- tion factor f. Embodied in all of the momentum analyses is a friction factor that relates the velocity to the bed shear stress and represents an empirical coefficient. This equa- tion was subsequently revised by Inman and Quinn (52) who found that a better fit to the data originally collected b/ Putnam et al . , and additional field data collected by the authors, was obtained if the constant friction coefficient xn the original equation was changed to be a function of t le velocity. ^33 tn (0 H 3 £ o tM c 3 u ?^ o CO bo c o CO <N 0) .-I n3 C o -H m w CO tn •H •H CO CO CO •H >, CO H c T3 to >i CO ro o 0) c (U H CO •H 0) C •H CO to CO CO to CO to to -H to to C >1 to to e lO rQ e H 3 1 1 to 1 3 3 H +j .H C M 4-' B to c ifl ^ to to to G U O <u O !m o O (U o •H s •H 0) -H 0) ■H e o u C u 0 g •H •H to e a, nb e c e c e e c to o o tj to [J o c !^ U 3 a isi O 3 CO bO CO C CO o o J u o c < to tu CO o 3 < CN in C •H CO cn CM 3 O CO c •w CO o a •H CO o CO CO to 3 •H J2 a< S to I fn 0) C pa CO CM a O 1^ c a o (a co C •H to O 4- C •H o H CO to D. S I (N o 3 H r- CO o o o 1 CQ + C to Em 4-' <X> CD rH O CO CO O o O o + +■ • O o CO •H u o H # CM ,^3 a a •H CO CO CM CM CM If) g o- I § i M 8 CO o o rQ CO. C to CO * CO o H 3& c o -H • +J O w o •H -M nj a I > -p c 0) C 3 O CJ Csl (U 4h O P w bO w C O J c (0 0) o o < H to o •H U •H g I •H 6 0) ca M o o Xi 3 C •H m CN o •H ■P > u to c o CJ m w n3 s a CM C ■H CO CO. C m +-> TO H c o E (U CO B o o to s 8 (M C •H CO CO. C m CM 0} X) " 3 G H O O •H C -P -H <TS > CO U +J OJ G CO a) G ^^ O CD 3 o CO CO CU CO Xi a CO o o 00. G CO ■p H CO 0) " 3 G H O C ■H ■P CO > u CD CO CD c u o 3 o CO CO O4 (0 '-H 3 c •H CO s CO o o c CO ■p CO H, CN CO a CN c •H CO OX +J C 0 G •H 0 ■P •H CO -P > CO U > CO tu CO CO 0) m > G > CO (0 0 lO G S 0 a 0 >, s >i u 3 U >> CO 4-' CO bjQ 4-' G ■P U -H CU ■H (U H B H C 0 0 0 U C/D X\ a c •H CO X tiO 00 CN CN a CO o o CO. G CO ■P CO CO o CN fa < |CN ■X3 0 3' .5 § O era 0 H CO U o ■p 3 O 00 CO G lO S G U to e 10 G O CO 0) H bO to w i G •H > to C5 CO in I 3 CO C 3 I 3 ,G O o T3 O G bfl lO P3 I G to 6 G M 0 H i i a •p 3 O to I a; G 3 I E to G 4-> 3 PL, CD H CM m m > c s o •rH O -(J -H > O P -H 0) fj M (U C CU O a e c <D e o I u •H u -H s E >, CO < in in C o w H bO ITS w CM CO -H CO CO CO c o •H (0 < e c s o PS s a; U 0) s c CM 3 D o c O iSl •H C CO •rH (D a CO !^ a CM IM O ;3 c 1 •H •H CO CO CO CO ^ — ' ba CO al CN c CO C . V o o •H J u CO CO 4- o c < /3 CN co V Me bO CO CO t= CD LO a- o 03 o CM CO CO -H CO CO <u u CO C o •H to ■H to _ Pi S C < ■p c 0) e o a CO o o a c •w CO a c to 4-1 iiclM-( to 03 c •H tin m •H a: I <u 3 § h3 36 Eagleson (55), usi.iig.the same cojitroJ. volume approach, developed a mathematical model to represent the growth of a longshore current downstream of a barrier,. In the associated laboratory experiments , it was found that a large percentage of the fluid composing plunging breakers (most common labora- tory breaker type) is extracted from the surf zone. This fluid already has a longshore velocity to which is added the longshore component of the breaking wave. This argument provides a mechanism for growth of the longshore current downstream. The asymptotic solution to the differential equation showed that the system is stable and that the growth of the currents was bounded. These results agreed qualitatively with laboratory results and demonstrated that unless there are perturbations inducing gradients in the wave energy in the longshore direction, the current system tends to be uniform alongshore and stable for stationary wave conditions . Bowen (M-7) used a conservation of momentum approach to determine the longshore velocity distribution across the surf zone for the case of a plane beach with a constant lateral shear stress. Reasonable results were obtained when compared to laboratory data. In the case of no lateral shear stress, his results are comparable to those of Longuet- Higgins except for inclusion of set-up. 37 Longuet-Higgins ' (-46 independently of -Bowen, derived a longshore velocity distribution across the surf zone for the case of a plane beach based on "radiation stress" theory. Reasonable results are also found for this theory when lateral mixing effects are included. As "radiation stress" theory will be used in the development of a sand transport equation, this model will be expanded on later. In -the- late 1960's Galvin^(56) critically reviewed the mean longshore current theories and tested those that predicted longshore current distribution across the surf zone as well as mean longshore current agreement. Galvin used both latoratory as well as field data. Galvin concluded that the most applicable equation appeared to be that of Longuet-Higgins (M-S) when proper unknown constant coeffi- cients were used. if Sonu et al. (49) 'Conducted field experiments and found poor correlation when compared to six of the above equations. Their experiments did point out the importance of the near- shore topography on the current system, and how this may affect the outcome of such results. Shepard and Sayner (57) pointed out that another reason for lack of agreement between mean longshore current' theory and field experiments could be the variation of current across the surf zone; each field data point is usually based on only a single location in the surf zone and additionally consists of effects based on a spectrum of waves rather than one frequency component. 38 Many simplifying assumptions are necessary in developing the theories. Since exact expressions are not available, it is necessary, inside the surf zone, to select approximate expressions for the wave speed, wave shape, water particle velocity, partitioning of energy in the wave field, a long- shore variation of waves and currents , and velocity and energy distributions across the surf zone. It is possible that -improved theories for longshore currents will require a better understanding of the highly nonlinear waves in the vicinity of the surf zone. The importance of considering factors other than wave parameters, such as the wind, was demonstrated by Sonu's empirical correlations. He found the wind to be the second most important variable in his set of field data. This shows the difficulty in comparing tests, particularly field data where information concerning the effects of bottom topography and winds often is not included. The extrapola- tion of data from one particular location to another without accounting for the importance of these effects can lead to invalid results. All of the equations involve unknown coefficients to be determined experimentally. Generally, the friction factor in the momentum and energy equations is evaluated in the same manner as in open channel hydraulics. The validity of utilizing results from steady flow situations without inertia 39 considerations certainly needs to be investigated further and could hopefully result in a refinement. Sonu points out that another possible improvement might derive from consideration of the dynamic processes of energy dissipation in the surf zone environment. The difficulty is that the flux of energy used in generating longshore currents is only a small fraction of the total available energy, and, as such, represents a second ord.er phenomenon (56). A difficulty with using the continuity approach is that, although it allows a description of the mean current, no procedure has yet been developed, based on continuity consid- erations, which can provide a prediction of the variation of the current across the surf zone. Momentum approaches remain the most useful approaches to date (55) when dealing with longshore currents and will be used in the development of a sand transport model. 2 . 3 Review of Friction Factor The longshore current in many of the above formulations is dependent on bed shear stress which in turn is dependent on bottom friction factor f ' . The determination of the bed w shear stress and hence friction factor for uniform steady flow has been fairly well established. For oscillatory flow, and particularly combined waves and currents, the bed shear stress is not well formulated. This is due primarily to the lack of good data. 40 Both -field and laboratory. data exist for the evalua- tion of the friction factor "f" although field data have been obtained on an "energy" type approach whereas most laboratory results have been attained by other means. As the two approaches provide somewhat different answers both will be discussed. 2.3.1 Field Data Field data for friction factor f ' have been obtained w by measurement of wave heights at two different locations and by calculating the energy loss due to bottom energy dissipation by friction, then equating theoretical energy dissipation to measured energy dissipation. The friction factor relates the shear stress per unit area at the bed T, to the horizontal flow velocity above the bed u, as h hw ^h = 1 ^^;p%w' ^2.21) other equations of the general form of equation 2.21 have appeared with the constant 1/2 replaced by 1 or 1/8. The rate of wave energy dissipation per unit area, E^, following Putnam and Johnson (58) is ^f ^h ^hw ^2.22) Combining equations 2.21 and 2.22 yields: ^f \ P%w' ^2.23) 1+1 All variables in the equations presented thus far indicate instantaneous values. For the sinusoidal flow at the bed, predicted by small amplitude theory. ^hw = ^hwm • (2.24) where u^^^ is the maximum horizontal bottom particle velocity at the upper limit of the boundary layer (i.e., the maximum bottom velocity of inviscid f low) . The argument of the sine term includes the angular frequency, w = 2Tr/T, with T the wave period and the time, t. Substituting equation 2.214 in- to 2.2 3 and integrating over the wave period gives, ^f = It P %wm^ • (2.25) Equation 2.25 is the average rate of wave energy dissipation per unit area of ocean bed due to the frictional shear stresses produced at the bed. Bretschneider and Reid (59) give a solution to the wave energy conservation equation presented in the form of a wave height reduction factor, K^. This factor accounts for bottom friction losses in a way similar to linear wave the- ory refraction and shoaling coefficient treatment of wave energy transformation. Combined they show the change in wave height due to three effects: H = ^s ^o • (2.26) The coefficients of K^, K^, and are the friction, refrac- tion and shoaling coefficients. 42 Br.etschneider (6 0) .using Jield data from wage gages on two oil platforms in the Gulf of Mexico calculated fric- tion factors averaging 0.10 6 by first calculating the factor from field measurements and then comparing the so obtained with the postulated energy dissipation using the Bretschneider and Reid (59) solution to obtain f. w In analyzing bottom losses for waves generated in shallow water, Bretschneider (5Q) found an average friction factor of f^ = 0.02. Iwagaki and Kakinuma (51) used signifi- cant wave heights at two stations off the coast of Akita, with the Bretschneider and Reid (50) method, to estimate friction factors ranging from 0.066 to 0.180, with an average of 0.114. Iwagaki and Kakinuma (62) found average friction factors for 4 other coasts of 0.280, 0.130, 0.110 and 0.100. The results of various field tests (60-63) are given in Table 2.4, which lists average f values as well as the ' ^ w range of friction factors measured " in the study. ■ Unfortu- nately, the results of field tests for friction factors are seen to vary widely which is due to a number of reasons: (1) additional wind energy effects on wave height in the measured values of wave height (which could increase or decrease wave height depending on whether wave energy was added or extracted from waves by wind between measuring points); (2) wave frictional damping due to viscid bottom fluid such as encountered on mud or fine silt bottoms; 1+3 Table 2.i+. Results of Field Measurements of Friction Factor. Reference Location f Average w Range of f ' Bret Schneider (50) Gulf of Mexico 0.106 0. 060-1. 93U Bretschneider (60) Iwagaki and Kakinuma (61-D2) Kishi (63) Gulf of Mexico Akita Coast, Japan Izumisano Coast Hiezu Coast (1963) Nishikinohama Coast Hiezu Coast (196i4) Takahama Coast depths 20-M-O ft. slopes - 0.006 Niigata , Japan depth 6-8 ft. slope 0.018 0.02 0.116 0.280 0.156 1.100 0.094 0.100 0.035 0.066-0,180 0.054-0.260 0.560-2.320 0.020-0. mo 0.060-0.160 0.03 -0.04 41+ (3) incorrect measurement, .refraction between measuring points; (4) no dependence on bottom roughness. The second reason for field data scatter has been mentioned by Bretschneider and Reid (50) as causing friction factors an order of magnitude larger than might be experi- enced otherwise in areas with mud bottoms. The fourth reason mentioned, that bottom roughness was not taken into- account is contrary to what one would expect as shown in steady flow friction factors from pipe flow data (64). 2.3.2 Laboratory Data Jonsson (65) compiled available laboratory data and conducted additional experiments dealing with turbulent boundary layers in oscillatory flow. From this study, he developed a classification of the flow regimes similar to that for steady flow. The classification is based on a roughness parameter and characteristic Reynolds number. The wave boundary layer in nature can always be considered in the "Hydraulically rough" turbulent regime according to this classification scheme. ' Unlike the boundary layer in open channel flow, which essentially extends over the entire depth of flow, the boundary layer under wave motion constitutes only a very small fraction of the vertical velocity distribution. This is because the boundary layer does not have an opportunity 45 to develop under the unsteady velocity field of the wave action. Above the boundary layer, the free stream is well described by the potential flow theory (for small waves). Jonsson using an oscillating flow tunnel to simulate prototype wave conditions in the laboratory was able to measure the vertical velocity profile in the boundary layer and determine the bottom shear stress. He found that for simple harmonic wave motion .the velordty profile in the boundary layer could be approximated by a logarithmic dis- tribution and that the instantaneous bottom shear stress was related to the velocity by the quadratic shear stress formula ■^h = ^w f ^hl^hl ^2.27) or ^h = ^w f ^hwm ^°s(a3t-8) | coscot-S | (2.28) where ^-^vim '^'^^ velocity amplitude of the oscillating flow just above the boundary layer, 6 is a phase lag, and f is a w friction factor associated with the wave motion. Jonsson also found that the friction factor was practi- cally constant over an oscillation period. The constancy of the friction factor for particular flow conditions is an im- portant result which allows for a better determination of the combined shear stress due to waves and the longshore current. Using the available data from several sources, '.■Jonsson found ■46 that the wave friction factor for laminar wave boundary layer flow was f„ = — (2.29) ew or, in terms of the friction coefficient as defined in the energy dissipation approach used in the field studies, is f; = (2.30) W /R ew where R^^ = wave bottom amplitude Reynold's number = . (2.31) For the case of rough turbulent flow Jonsson found that the friction factor for simple harmonic wave motion could be represented by an equation of the form 1 . 1 _ „ , . ^h _ + log — — = -0.08 + log — ~ (2.32) 4 /f 4 Vf~ ^ w w where r is a measure of roughness, and E,^ is the maximum horizontal water particle excursion amplitude of the fluid motion at the bottom as predicted by linear wave theory ^ 2 sinh kh (2.33) Equation 2.3 2 is based on the roughness parameter r being equivalent to the Nikuradse roughness parameter (65). As the roughness is a function of ripple height and spacing in wave flow, f is also a function of the wave w 1+7 characteristics. This is because, for the granular beds consisting of a particular grain size, the ripples adjust their dimensions according to the wave motion, and it is the ripple geometry that determines the effective roughness. Using additional laboratory data, others (66-69) have postulated results similar to Jonsson where the entire range of wave friction factor f depends on both the wave bottom w amplitude Reynold's number R^^ and boundary roughness — wave amplitude ratio. A summary of some laboratory results for the rough turbulent range are given in Table 2.5. Here it is important to again note that f^ and f^ are not necessarily equivalent as wave friction factor f^ has been arrived at assuming no phase lag between bottom wave stress and wave velocity while Jonsson 's data include a phase lag between maximum bottom wave shear and maximum bottom wave velocity. . Also, Jonsson 's postulated friction factor f takes into account '' ■^ bed roughness effects w whereas the data used in estimating the friction factor f^ did not account for bed roughness form explicitly. At present there is only a qualitative understanding of the ripple geometry as related to sand wave characteris- tics. Generally, the ripples are much more symmetrical in shape and much longer crested as compared to those found in alluvial channels. Inman (70) collected a large number of observations of ripple geometry and wave conditions from 48 Table 2.5. Results of Laboratory Measurements of Friction Factor. Tp=!t Pnnditions f w Jonsson Oscillating water channel 0. 02-0. 30 \03 } LLlxDUJ-cii L ijtju.iiu.cix _y xci_y t3X additional reanalysis of other laboratory experiments Tunstall Wave flume study 0. 09-0. 50 and Inman U, T 2 t; R Iwagaki Wave Flume Study 0. 01-0. 40 and. laminar boundarv laver Tsuchiva (67) Kamphuis, J. W. Similar to Jonsson 0. 02-0. 30 (68) experiments Car St ens, et al. Oscillating flow in channel. 0. 01-0. 50 (69) turbulent boundary layer R = Ripple height. 49 Southern Calif orBia .beacJies - T±i£se ..otiser'vations extended from a depth of 170 feet to the shore. Since the wave and sand characteristics vary from deep water to the beach, the ripple geometry would be expected to vary also. These ob- servations showed that the size of the sand is the most important factor in determining the geometry of the ripple. In general, the coarser the sand, the larger the ripples. Also, there was a general correspondence of decreasing ripple height with decreasing water depth. The ripples were smallest in the surf zone where the higher orbital velocities of the waves tended to plane the ripples off ; the ripples were almost nonexistent for surf zones with fine sands. The ripple wave length was related to the wave bottom orbital excursion. As the bottom orbital velocity of the shoaling waves increased, the ripple length decreased, increasing the effective roughness, but, at the same time, the ripple height is decreased, decreasing the effective roughness. Carstens et al. (69) in oscillatory flow laboratory experiments found that the ratio of ripple amplitude to mean sediment diameter and the ratio of dune amplitude to dune wave length were found to be unique functions of a single variable — ratio of water motion amplitude to mean sediment diameter. If the amplitude-to-diameter ratio is less than 775, the ripples are two dimensional with essentially straight and level crests and troughs. If the amplitude-to- diameter ratio is greater than 1700, the bed is flat 50 regardless of the initial condition. On the intermediate range from 775 to 1700, the ripples are three-dimensional with ill defined crests and troughs. Ripple amplitude was found to decrease almost linearly with increasing water motion amplitude in the range of three dimensional ripples. CHAPTER III REVIEW OF EXISTING DATA ON LONGSHORE SAND TRANSPORT 3 . 1 Introductory Remarks Existing littoral sand transport data fall into two categories, laboratory data and field data. Data on sand transport and waves are diffiault to take under laboratory conditions, not to mention the inherent difficulties in taking field data. The following review of available data on sand transport includes manner of testing where available data exist, and conditions of such testing. The review is comprehensive to the best of the author's knowledge, although the data are by no means ideal. In almost all tests of sand transport many important parameters have not been measured and very rarely has differentiation been made between bed load transport and suspended load transport. Much of the data herein referred to have been used in past studies for analysis of sand transport models although only order of magnitude agreement has been found to date. Many of the laboratory and field tests included here have also been reviewed by Das (71). 3 . 2 Laboratory Studies A number of laboratory experiments have been conducted to determine the mechanisms causing sand transport in the 51 52 surf zone. This discussion is limited to the three- dimensional studies simulating conditions in the prototype. Krumbein (72) conducted one of the first of these experiments and concluded that the mean littoral drift was a function of the deep water wave steepness, H^/L^. Subsequent studies by Saville (73) showed that a maximum transport occurred for a wave steepness of 0.025, and, for steepnesses greater or less than this value, the transport was less. Saville also found that under low wave steepness much of the sand trans- port occurred on the foreshore of the beach as bed load while for higher steepness waves much of the sand transport occurred as suspended load. Both Saville (73) and Savage (7M-) concluded that greater than 9 0% of all longshore sand transport due to waves occurred within the surf zone (land- ward of breaking waves). This has also been confirmed by at least one researcher in field studies (75). Although laboratory results imply that the transport is a function of wave steepness, the reason for this dependence has not been established. Galvin (76) conducted a series of experiments of breaking waves on laboratory beaches . He developed a classification for determining whether the waves develop into plunging or spilling breakers as related to wave steepness and the beach slope. These results, when compared to the littoral drift studies in the laboratory, indicate that maximum transport occurs for a plunging-type 53 breaker and that the rate of. transport may be more a func- tion of the manner in which the waves break than the wave steepness. Battjes (7 7) has defined the parameter which controls the type of wave breaking as a surf similarity parameter, c, = "^ ^"^ , where B is beach slope angle, H is o wave height, and is deepwater wave length. The smaller the surf similarity parameter the more the wave breaking process approaches spilling breaker conditions. The following Table 3.1 provides a review of laboratory experiments which will be referred to in later discussion. Table 3.2 gives the range of wave basins used. The angles are the nominal settings of the wave generators to the beach. In the case of "snake" wave generators, angle is controlled by wave phasing. A short review of each series of tests follows: 3.2.1 Krumbein Laboratory Study (72) The laboratory study by Krumbein was conducted in 19142 at the University of California, Berkeley. Longshore trans- port rate, longshore current velocity, and wave and sediment characteristics were measured. The maximum rate of removal of sand from the updrift end of the beach was determined from the rate of feed of material to a hopper. The rate was adjusted to the capability of the waves to move sand. Longshore currents were measured by using floats and confetti. O M +J M • 0) O E-i f4 O +-> u a a > to •H -P •H C l-l 0) o rH ban a <u <u o u bO <D c o < ^ -p a d) 0) CO O ■P -P •H ip 0) M C O o 0) O •H p c ra o ji, p CO ■H ., •H <D t3 -P oj o e CO rO g to O a to Q o <n o u 3 O to to O • o o CO O rH o rH H O O O rH <M O CO rH H T3 !>. <D ■P H OJ <U C •H rH rH rH 0) j:: <U a o o — , o A:: o c c u f< ?^ 03 0) •H a; •H 0) 0) C Sh > w to m m pq o Jo pu >, >1 >, to >) c u to O -H O o G o o +-> 0) ■P < — i ■p to p ■p 0) to to rH to rO to to bO !^ S u u >, U > 1^ to O 3 o > o to O o > IB ^ to ^ x: to X) to IB to (B CO to CO J J o CO CN O CO O I in o o o J- o CO CD CO rH in 00 jj- CO CO CN H r-i rH CN H rH rH r- o CO rH CO in CN rH CN rH CN CN CN 1 CD 1 cn 1 CN 1 in 1 03 1 CO 1 CO r~- in CO H rH o H o o o o o =f o o in CO CO ID o in J- cr> H CN 1 rH H CO H r- 1 =t- 1 lO in 1 in 1 1 <T> H f- CN H o cn rH rH rH rH in Ol <J> o o o in in in lO to rH CO CO (O CD CN CN CN rH H rH rH OJ o o o o O O CN o o in CO CO in in o CO CO c^- o o o o rH rH o o H y — s. 00 CD cn o rH Ol CO CO T3 CO CO CO bO - C C •H • . o P" H Xi Mh rH CO H IB ■ ■H OJ S rH J Q — ' >, !^ O QJ o o O P p H Pi U (0 (U to CU IB 0) U O t< O A< 0 -H o 0 •n !^ Xi rO Xi •rH to to cq to cq -3 55 Table 3.2. Summary of Wave Basin Dimensions. Source of Data Krimbein Saville Shay and Johnson Sauvage and Vincent Fairchild (CERC) and Savage Price and Tomlinson Barcelo ' Bijker Basin Dimensions (feet) 58.2 X 38.7 X 2.0 122 X 66 X 2.0 122 X 56 X 2.0 Not Available 150 X 100 X 30 (North Sector) 200 X 150 X 30 (South Sector) 190 X 75 64 X 32 Approximate Beach Length (feet) 38 60 60 (Counts 25-81) 33 (Counts 82-123) Not Available 30-98 (Variable) ? 32 ? 56 The maximum velocity parafllei to shoi^e near the plunge line was measured by releasing a soaked string, and recording the movement of the end of the string over a fixed time interval. Wave heights were measured with a combination point and hook gage in which the hook-could be set to the troughs of the waves and the point to the. crests. The wave height was then obtained by reading the difference in gage heights on a single vernier. The mechanical analysis of sand used in the study gave a median grain diameter of 0.5 mm, although the sand was not well sorted. Specific gravity of material (sand) was assumed equal to 2.65. Unfortunately wave breaker angles were not recorded. 3.2.2 Saville Laboratory Study (73) Saville conducted a laboratory study at the University of California, Berkeley. The total transport rate due to bed load and suspended load was determined by installing a weighing device at the downcast end of the beach. The bed load rate was measured by installing hoppers on the beach. Breaker angles were determined from vertical photographs . The mechanical analysis of sand showed a median gran size dgg = 0.30 mm. The sand was relatively uniform in size with a specific gravity of 2.69. 3.2.3 Shay and Johnson Laboratory Study (78) A laboratory study was conducted by Shay and Johnson at the University of California. VJave and sediment 57 characteristics, longshore transport rate, and maximum longshore current were measured in the tests. Wave height variability was observed in the experiments therefore many readings were taken during each run and an arithmetic average of the readings was used to characterize the wave height, ^ave heights were measured with a point and hook gage as in Krumbein's tests. The bed load rate was measured by install- ing hoppers on the beach for small generator angles. These hoppers were removed when wave splitters were used for angles of 3 0 degrees or greater. Both Bed'load and total load were measured in these tests similar to those of Saville Longshore current velocities (maximum and the average) were measured by releasing fluorescein. The breaker angles were determined from vertical photographs as in Saville 's study . Equilibrium beach profiles were taken after a stable condition was reached by inserting a piece of sheet metal in the beach perpendicular to the beach contours and tracing the sand profile under water with a grease marker, with a view to study the influence of wave steepness on beach profile . The median grain size was d^^ = 0.30 mm and specific gravity was 2.69. 58 3.2.4 Sauvage and Vincent.- T. a, boj-a-tory Study (79) Two series of laboratory experiments were conducted by the authors at the hydraulics laboratory at Grenoble, France. In the first series of tests, the beach was set at an angle of 15 degrees with the wave makers. In the second series, the angle was varied from 5 degrees to 70 degrees. In addition, there was a unique set of experiments in which three types of sediments with different characteristics were used. The volume of material, fed from a distributor and transported along the beach, was collected in a trap at the downdrift end. The data from the measurements of longshore transport and the wave characteristics are presented ' only in graphical form in the paper by Sauvage and Vincent (79). The actual measured data are not available. In the studies by Sauvage and Vincent (79), the mean grain sizes of the three materials used were 0.5 mm, 1.5 mm, and 1.0 mm, and their respective specific gravities were given as 2.60, 1.40, and 1.1. 3.2.5 Savage and Fairchild Laboratory Study (74, 80) In a laboratory program on the studies of longshore transport at the former Beach Erosion Board (BEB), presently known as Coastal Engineering Research Center (CERC), several tests were conducted from 19 5 8 to 1965. Ten of these tests were reported by Savage (74), and all of the tests in edited form appear in Fairchild (80). 59 The transport studies were made in the north and south sectors of the Shore Processes Test Basin (SPTB) of CERC. A detailed description of the SPTB, the sand transporting system, the sand traps and the sand weighing system is ..in BEB Technical Memorandum No. 11M-, 19 59. The total quantity of longshore drift caught in the sand traps was recorded for every 5-hour interval during the test. To establish a r e la tionsiip . between the longshore energy and the longshore transport rate, the transport rate between the 20th and 30th hours of test was used, whenever the test continued over 20 hours. Due to the large variability in wave heights observed in the SPTB during the test, it was difficult to characterize each test with a particular measured incident wave height. It was decided to characterize the waves by a half-size Froude model in another wave flume (75 feet by 1.5 feet by 2.0 foot deep) at CERC. By varying the eccentricity and the period in the wave flume, the wave heights (before visible reflection occurred) were measured. The average of these wave heights was considered as the height of incident wave at that particular depth of water. The wave height and eccentricity were derived from half-scale, Froude-model wave tank studies. A family of curves was obtained for wave heights as a function of wave period with eccentricity as a parameter. 60 These curves were used to diajpacterize the incident waves in the SPTB (8), and these values (due to the large vari- ability in wave heights in the measurements of wave heights in SPTB) were used to compute the wave energy in the SPTB. Fluorescein was used in most tests to measure longshore currents. Beach samples were also obtained in the tests.' Water temperature measurements were made during each test at frequent intervals. In most of the studies conducted at BEB, no training walls were placed in the wave basin to conform to the wave orthogonal for uniform distribution of energy in the beach. The BEB study also differs from other studies in such aspects as basin geometry, beach slope, wave height and period variability, and sand feeder height with respect to still- water level. Median grain size. was d^^ = 0.22 mm and specific gravity 2.65 (beach sand). The edited data appearing in Reference (80) are the results of the averages of various tests . 3.2.6 Price, Tomlinson, and Willis Laboratory Study (81) Studies were conducted at Wallingford Research Station (England) on the effect of groins on a beach stable for a particular wave condition and beach material. Longshore transport was also measured on two occasions without groins. The wave basin is 190 feet by 75 feet equipped with a snake type wave maker, and tide and tidal-current generators. 61 The initial beach slope was. 1 on 17 .at .mean tide level. The mean grain size and the specific gravity of crushed coal used in Price, Tomlinson and Willis's studies (81) were 0.80 mm and 1.35 respectively. 3.2.7 Barcelo Laboratory Study (82) Studies were conducted at the Portugese National Engi- neering Laboratory on the effect of groins on a beach for various wave conditions . Longshore transport was measured in the first hour of the tests by measurement of accretion volume at a long groin within the basin. Wave heights measured were an average of random wave heights which- cor- responded to a Gaussian distribution. Median grain size was 1.30 mm and specific gravity equal to 1.6 7 for pumice stone, the material used in testing. As data were not listed in (82) except in graphical form, wave energy values and sand transport values have to be taken off graphical plots in the report. 3.2.8 Bijker Laboratory Study (83) The raw data presented in the Bijker (83) report are sketchy and considerable questionable aspects of the study are not answered. The longshore current induced in the model is not driven by waves alone but by a combination of waves and an imposed gravity slope in the longshore direction. As a result, the data do not pertain to wave driven sand 62 transport alone but must be the result of additional forcing functions. 3 . 3 Field Studies The measurement of littoral drift along a particular beach and the meaningful correlation with the wave properties are an extremely difficult field task. Thus, only limited field data are available. The earliest published information on the distribution of sand transport along the beach profile for prototype conditions was by the Beach Erosion Board (SH). Unfortu- nately, comprehensive wave data wezre not obtained. The distribution of longshore currents and suspended sand, obtained from water samples, was measured from piers extend- ing across the surf zone. These measurements showed that the greatest sand transport occurred at the breaker line, where the turbulence was a maximum, and decreased shoreward with another peak in the swash zone — another area of high turbulence. Seaward of the breakers the sand transport decreased with increasing depth. There have been few other field experiments of this type. Improvements in tracer techniques, particularly using either radio-active or fluorescent sand tracers, have in- creased the intensity of littoral drift studies in the field. It is generally assumed that the fluorescent tracer moves as bed load. A great number of studies have been 63 conducted in recent years on fluorescent tracing and have been sununarized in a book by Ingle (75). Ingle also con- ducted a number of studies using fluorescent tracers on several Southern California Beaches. The results of these studies and some studies conducted by Thorton (30), and also earlier investigators, are all generally similar to that given by Zenkovitch (85) and shown in Figure 3.1, describing the variation of sand transport across the surf zone. The fluorescent sand grains are found in greatest concentrations along points of high turbulence such as wave breaking areas . In a bar- trough profile, the sand moves predominantly along the bar or in the swash zone. There is a minimum of tracer transport in the trough. During the tests shown in Figure 3.1 the waves were relatively high and spilling breakers prevailed . Tracer studies have been conducted using both the Eulerian and Lagrangian approaches. Most investigators have used the Lagrangian approach in which the tracer is intro- duced at a particular location, and the concentration dis- tribution of the tracer is determined by obtaining samples at various sample points. The concentration of tracers is then determined by counting the tracer grains in the sand samples. The Eulerian approach is to sample in time along a particular line across the surf zone, traversing the path of the tracers. A stable platform or other work facility is 64 Outer Breaker Line Q 28 4 2 0 75 (m) ICO 150^-4- X i— a. UJ Figure 3.1. Distribution of Longshore Velocity and Sediment Transport Across the Surf Zone (after Zenkovitch (85")) 8 5 generally required in this method. This was the approach used by Zenkovitch (85), working from a tramway traversing the surf zone. Bruun and Battjes (86) and Thornton (30) worked from a pier. The inherent difficulty of fluorescent tracer studies is that quantitative measurements require recovery of most of the tracer. Unfortunately, the recovery level is gener- ally very low, amounting to only. a few percent. This re- quires that accompanying measurements of the quantity of sand in suspension, or moving on the bed, also be determined. However, as a tool, or aid, for solving engineering problems in which qualitative information can be extremely important, fluorescent tracer techniques alone can be of great value. Due to lack of field data such studies are necessary for a large scale working sand transport correlation until much more field data are obtained. Table 3.3 presents a summary of field tests in which measurements were made of both sand transport and wave parameters. None of the field tests have differentiated between bed load and suspended load. A short summary of each of the various field tests follows . 3.3.1 Watt Field Study (87) By measuring wave characteristics, longshore currents, and amount of material pumped by the bypassing plang on the 66 m >, c bO o c u > m :2 3 o rd a CO o 1 t +-* Q J— M rH CD ■H bO U !^ 0) bO ^ a; 0) bO M -P (11 CO bo fij O 0) t3 > C Cu m 1 1 E- S 1 1 n CO (0 H <l) CO CO to on <u 0) •H ■p p p P P CO •H •H •H 10 •H tu CO CO > u (7D -3-1- T) T) T3 T3 6 CU <U CL) CO tu •H P P P P P C fl C o C Cd 0) 0) (U <1) B s rH s 3 to 3 H u ?j to p P P CO P 3 CO CO to •H to to In In In > In Vi 3 C7< p O CO c to CO E-< •H c o CO c .H to to C/0 < c o •H P to u o bO c •H CO bO to to >, o ^ p •H T) C C O to S P to c o •H to O 0) c o •H P (U u O 0) O T-l t p CO (U O to CO p ip c o me (1) tu 0) <u p CO H (0 s 3 ft CO to C to u •H fO 3 <U 6 p P C T3 ? CO o to o •w to o p tu O bO ID H o o t3 P to O •H o CO 0) ^ ft < bo an e o tn P CO 0) E-> rH •H O in in CD CD CN CM CN O o o CD o O O to c y — ^ O CD tn cn ip CD CO ■H (0 ^ — ' rH •H p to to x: G O o o -H to O 3: ip O eq •H 0) rH to H m iH O to lo rH C O e cu 0) p CO •H x: (U P o; X) 0 to to p H P ,a .H •i—i e 3 C to 0) to lO o o IH c O CO < M LO CD in <D Ln iH o CO to CO •H w c U o CD CO in CD O in CD o CO o o to COo lOtxi t3 ip (U c •H CQ O to H tu CO to e (0 ft P O <u C C e CO c •H to O o 3 13 P U x: C ■H c £-< a (0 a to u > to 6 P c O o <U H 1) O !^ U u rH ft •H eq O CD 0) tM E-r 10 CO Oh o 67 north side of the north jetty at South Lake Worth Inlet, Florida, Watts attempted to relate the volume of longshore transport reaching the pump intake to the wave energy reach- ing adjacent shores. (87). In the South Lake Worth Inlet region of the Florida coast, net longshore transport is from north to south. The material intercepted by the north jetty of the inlet was bypassed froni the north side to the south shore of the south jetty to prevent shoaling inside the inlet and to nourish the south beach. The southerly component of longshore transport near the inlet was measured from a detention basin, which was prepared for this study. The sand was pumped into this basin. During the period sand could not be pumped into the basin, the pumping rate of the plant was used to estimate the material pumped. During the period of study, the pumping plant bypassed almost all the littoral drift moving alongshore inside the surf zone. Therefore, it was assumed that the pumped volume would represent the total southerly longshore transport rate in the nearshore zone. The material pumped into the deten- tion basin was periodically surveyed for measurement of the quantity. During the period the detention basin was being cleared and leveled, the material could not be pumped into the basin, but the material pumped during these intervals was estimated from the average pumping rate of 7 6.2 cubic 68 yards per hour, computed from the log of pumping time between January 1949 and December 1951, furnished by the Palm Beach County Engineer. Wave heights and periods were measured by a pressure gage located at Palm Beach Pier, 11 miles north of the inlet and in about 17 feet of water below mean sea level. The recording mechanism was programmed to obtain a 12-minute record every 4 hours from 6 March to 10 June 19 52. Wave directions were measured twice daily by the use of a sighting bar and auxiliary sights attached to an ordinary engineer transit located on the roof of the Ambassador Hotel about 3.5 miles north of the bypassing plant. Significant wave heights and periods were computed from the wave records from a pressure gage in 17 feet of water. Frequencies of wave heights and periods were plotted. A wave direction frequency plot indicated that 7 5 percent of the direction were from north of east. The predominant direction of longshore transport along the Florida Coast is generally from north to south. Therefore, for the data presented in the analysis, the alongshore component:- of wave energy was computed for each month for southerly wave direc- tions (a<90°) from durations for each a and the corresponding recorded wave height and period data. The total monthly southerly longshore transport was evaluated from the pumped material. The southerly component of wave work was expressed in footpounds per day per foot of wave crest, and the 69 southerly rate of littoral transport wa.s expressed in cubic yards per day. Longshore currents inside the breaker zone were measured twice daily by using fluorescein at four locations at distances of 1/2 mile, 2 miles, 5 miles, and 7 miles north of the bypassing plant at the same time^-the wave directions were measured. Sediment 'Samples were taken during the study and size distributions were made by Emery Settling Velocity Tube. This material was a sand and shell mixture. The data for sampling stations 1/2 mile north of inlet and 1,000 feet south of inlet on different dates and at high, mean, and low tide lines, and at 3-foot water depth were presented. 3.3.2 Caldwell Field Study (88) From measurements of the rate of a longshore sand movement of a beach fill placed at Surf side, south of Anaheim Bay, California, and the associated wave charac- teristics, an attempt was made to correlate the two (88). The beach profile changes shown by seven surveys out to the 20-foot contour were used to compute the volume changes of beach fill along the shore. Wave heights and periods were obtained from 8-minute wave records taken at 4--hour intervals by wave gages in- stalled on the Huntington Beach Pier about 6 miles south of Anaheim Bay. The depth of water at the gaging station was 70 20 feet below mean lower low. water. . Utilizing meterological data, wave forecasting techniques were used to "hindcast" significant wave heights, significant periods, and the directions of wave approach. Hindcast heights and periods were used to supplement recorded heights and periods where gaps occurred in the record. Aerial photographs of the study area were also taken. Wave direction and period were also determined from the photographs. Hindcast ing and wave refrac- tion analysis were used in combination to determine the wave direction associated with each wave observation. The median grain size in the beach fill area was 0.42 mm. 3.3.3 CERC Field Study (89) A study was made by the Coastal Engineering Research Center of the sand deposited behind an offshore breakwater at Port Huneme, California (89). The sand deposition rate was estimated by means of detailed hydrographic surveys . Wave height, period, and direction data were taken by visual observation at the site and correlated with the sand deposited in the lee of the breakwater. Unfortunately visual observa- tions were not available throughout the entire length of the study. 3.3.U Moore and Cole Field Study (90) A transport rate of 4,6 80 cubic yards per day was measured near Cape Thompson Alaska, from the growth of a sand spit during a 3-hour period (90). The material was 71 deposited on the spit by waves 5..- feet high with a period of 5.5 seconds and a 25-degree angle to the beach. The' median grain size was 1.00 mm. 3.3.5 Komar Field Study (38) Studies were conducted at El Moreno Beach located on the northwest shore of Gulf of California, in Baja California, Mexico, and at Silver Strand Beach near San Diego, California (38). The purpose was to obtain field measurements of the bed load transport rate over short periods of time and, simultaneously, to measure the waves and currents to be able to test relationships between the longshore transport rate and the wave energy flux. The measurements of longshore transport rate of sand were made through use of natural sand colored with a thin coating of fluorescent dye. The sand advection rate was determined from the time history of the movement of the center of gravity of the sand tracer which had been intro- duced onto the beach. The thickness of the sand in motion was obtained from the depth of burial of the tracer sand in cores of the beach face. The product of the advection rate times the cross section of the sand in motion gave the sand transport rate. The wave direction and wave energy flux were obtained from simultaneous measurement-: of wave charac- teristics by an array of digital wave sensors placed in and near the surf zone. The wave sensors were of pressure type 72 measuring pressure variation near the bottom. The energy density obtained from measurements of the pressure trans- ducers was corrected for the damping effect due to over- lying depth of water using the linear pressure response factor. The root-mean square wave height obtained from the energy density, and the characteristic wave period obtained from the frequency spectrum were used to compute the wave energy flux at the breaker depth. Median grain size was drn = 0-6 mm at El Moreno Beach and 0.17 5 at Silver Strand 5 0 Beach, Specific gravity was 2.65. 3.3.6 Thornton Study (30) In this study bed load measurements were made of sand transport across the surf zone (30). The traps extended 8-10 inches off the bottom and were estimated to have an efficiency of 50-90%. The portion of suspended sand captured by the traps is unknown; therefore, it is questionable as to whether the entire amount of sand measured in these experiments is bed load or whether an important percentage is also suspended load. Wave measurements were made at an instrumented pier. CHAPTER IV SAND TRANSPORT MODEL DEVELOPMENT 4 . 1 Introduction Past investigations, as discussed in Chapter 2,' have attempted to relate the mean longshore currents to wave- induced momentum, energy, or. mas.S--£liix into the surf zone. The distribution of longshore currents can be similarly investigated by considering the changes in the momentum, energy, or mass flux across the surf zone as has been done by Longuet-Higgins (46), Bowen (47), and Thornton (30). The present analysis utilizes the momentum principle and the concept of "radiation stress," the excess momentum flux due to wave motion (91), to develop the working equations for longshore current and available power for sand transport. It is known that, due to the fluctuating water particle motion of the waves, there is a momentum flux component. If the waves have a direction component parallel to shore, a longshore current can be generated due to changes in the longshore momentum flux component of the shoaling waves (46). It is also known that there must be a displacement of the mean water surface elevation to balance the changes in the onshore momentum flux component of the shoaling waves (92) . 73 7^ In waves, the momentum flux is the sum of the pressure and the product of two velocities. It can be shown that the average momentum flux is nonlinear in wave height. In order to specify the excess momentum flux of the waves, it there- fore becomes necessary to consider nonlinear, or higher order effects of the wave motion. Since the waves are assumed in the present model to be the mechanism responsible for the generation of the longshore currents , much of the following discussion will be devoted to describing the waves . The treatment of periodic gravity waves is generally developed by perturbation schemes built on the exact solution for the linearized equations of motion with specified bound- ary conditions. Difficulty, with this type of analysis, is encountered when considering higher order theory with non- horizontal bottom boundary. For waves traveling in water of varying depth, two different approaches can be employed: the analytical method or the energy flux method. The analyt- ical method involves solving the boundary value problem accounting for the slope of the bottom to the desired degree of approximation. This technique includes the slope, explic- itly, in the perturbation expansion. The solution for the waves on a plane sloping bottom has been worked out to the second order by several authors (93). 75 The analytical method is generally more consistent because it accounts for the bottom slope in the bottom boundary condition to the same order of approximation as is maintained in the free surface boundary conditions. Small bottom slope and amplitude are assumed in the perturbation analysis. Although more attractive from an analytical point of view, this method has the inherent difficulty of requiring tedious computations for each.. particular case. Also, the rigorous boundary value problem approach is not readily extended across the surf zone. The energy method consists of solving the wave problem for a horizontal bottom and then extending these results to a sloping bottom by means of energy flux considerations. Hence, for short distances, in the region outside the surf zone, it is assumed that waves on a sloping bottom can be considered the same as on a horizontal bottom. Then, adja- cent increments of distance are connected together by means of the energy flux conservation equation. This results in a prediction of the wave height at any location outside the surf zone. Inside the surf zone, it will be assumed that the wave height is governed by the local water depth. The energy method allows for the inclusion of energy dissipative effects, such as bottom friction, and the focus- ing or spreading of energy due to refraction by changes in bathymetry and currents. These effects are believed to be 76 more important tlian the deformation of the water particle motion due to the bottom slope as given in the analytical procedure (93). The validity of the energy method can be proven by comparison with the results of the exact analytical solution. To the first order of approximation in wave energy, for the case of small bottom slopes, identical results are obtained for the propagation, of energy. This confirms the use of the energy flux equation to connect the solution for various depths for small bottom slopes, at least to the first order in wave energy. In the present approach, the energy method describes the wave field to the first order in energy (second order in wave amplitude). The solution to the energy equation is substituted into wave momentum conservation equations to obtain the working equations needed for predicting longshore current and hence sand transport. 4- . 2 Review of Mass, Momentum, and Energy Conservation P'eri'odic Wave Flow ' A starting point for the analysis is a review of the conservation equations of mass, momentum, and energy fluxes applicable to unsteady wave motion. The analysis is not concerned with the internal flow structure of the fluid; hence, the derivations can be simplified by integrating the conservation equations over depth. Conservation equations 77 which have already been developed by Phillips (94) will be used and are presented in a similar format to Phillips. The conservation equations will be applied to wave motion, but they are equally applicable to general turbulent motion. The unsteady velocity field of the wave motion can be expressed in the same manner as in the treatment of turbulent motion as the sum of its mean and fluctating parts - Resultant Mean Fluctuating u = U.(x,y,t) + ul (x,y,2,t) ,w(x,y,z,t) (4.1) where (1,2) refer to the horizontal coordinates (x,y), respectively, and z is the vertical coordinate. The tensor notation is used only for horizontal components of water particle motion. The mean current is assumed uniform over depth for simplicity. The pressure term can be stated similarly. These expressions can be substituted into the mass, momentum, and energy equations, and the mean and fluctuating contributions identified. The conservation equations are to be averaged over depth and time (one can consider averaging over a few wave periods). For the case of waves superposed on a mean cur- rent, all the wave motion would be identified with the fluctuating quantity which, when integrated over the total depth, can contain a mean contribution due to the waves. The time averaging of the equations for a general development, 78 being over a short interval compared to the total time, does not preclude long term unsteadiness in the mean motion. The utility of the conservation equations derived by Phillips is that the terms involving the mean and fluctuat- ing quantities have been separated. This facilitates the understanding of the effect of the unsteady wave motion on the total flow phenomenon. They are a particularly useful aid in gaining phy-s-ical . insight _ into the complicated mecha- nisms taking place in the surf zone. In addition to the conserved quantities of mass, momen- tum, and energy, a fourth conservation equation is available — conservation of phase (conservation of waves) (95). The conservation" of phase will be derived for simplic- ity in relation to simple harmonic motion. The expressions to be derived are also applicable to more general wave motion which can be formed by the superposition of individ- ual Fourier components. The expression for the water sur- face profile for harmonic motion is given by 1 = 1,2 (4.2) where j Cx^) is the amplitude, and the quantity Ck^x^ - cot) is called the phase function X. The wave-number k^ and frequency to can be defined in terms of the phase function. such that ■3-X C4 . 3 ) 79 An important property of the wave-number can be seen immedi- ately by vector identity ^xk=VxVX=0 (4.5) so that the wave-number is irrotational (see for example (94)). From equations 4.3 and 4 . M- , the kinematical conser- vation equation for the wave-number can be written 8k. „ 1 + 4iiL. = 0 (4.6) 3t 8x 1 For a single wave component, the conservation of phase says that the rate of increase of the number of waves in a fixed length is balanced by the net inward flux of waves per unit time. If (1) can be expressed as a function of k^ and pos- sibly X, from local arguments, (such as assuming a uniform simple harmonic wave train locally), and further allowing a mean current U, then, 0) = a(k,x. ) + k.U. (4.7) a is the local wave frequency and is the apparent frequency to an observer moving with the current. Substituting into Equation 4,6, Whitham (95) shows 3k. 3k. 3k. ^ + c . + U. ^-^ = - G. (4.8) 3t gj 3Xj : 3Xj 1 where „ 3U. G. = + k. ^ (4.9) 1 3x. 3 3x. 80. and c . = If- (^.10) c . is the speed at which the energy or values of k- are propagated, commonly called the group velocity. The individ- ual crests propagate with the local phase velocity c = ^. The notation shall be adopted where, if c or k appear with- out vector notation, it is understood that they represent the modulus of the vector quantity. Equation 4.8 says that the rate of change of the wave- number k^, following a point moving with the combined group and convective velocity, is equal to - G^. Changes in are due to variations in the mean current and bottom con- figuration. These equations, which state the kinematical conservation for the wave-number, hold for any kind of wave motion (Phillips (94)). The general conservation of total mass per unit area can be expressed |^pD.|_S. .0 i = 1,2 (^.11) i D is the total averaged depth of water (which can include a mean elevation "n, above (or below) the still depth h) so that D(x,y,t) = (n + h) (4.12) The overbar is used to signify a time average. The total mass flux can be partitioned into its mean and fluctuat- ing components 81 M. = M. + M. (H.13) 111 The mass flux per unit width of the mean flow from Phillips (94) is M. = 1 _^ pU^ dz = pDU^ (1+.14) The mass transport of the fluctuating wave motion is M. 1 n pu! dz ■ (4.15) -h ^ The equation defining the conservation of horizontal momentum is derived by integrating the momentum equation, including shear stresses, over depth and averaging in time. The balance of total momentum per unit area can be expressed as per Phillips (94) —M. + ^ (U.M. + S. . ) = F^. + R. (4.16) 3t 1 9Xj X : X3 px X Here M- denotes the total horizontal momentum per unit area. X Hence, the first term on the left represents the rate of change of the total mean momentum per unit area which in- cludes both the current momentum and wave momentum. The total mean transport velocity can be expressed in terms of the total mass flux per unit width as M. M. U. = = U. + (4.17) X pD X pD The second term on the left of Equation 4.16 expresses the momentum flux of a steady stream having the same mass flux 82 per unit width and mean .transport veJLojcity as the actual flow together with an excessive momentum flux term S^j arising from the superimposed wave motion, where ^ij ^ M.M. (pu!_u! + p6^j) dz - 1/2 PgD^fi^j - ■ (4.18) -h and is the Kronecker delta. This term is referred to as a "radiation stress" term by Longuet-Higgins and Stewart (91). The term F . is given by Fp. = - pg(n - h) II- (^.19) and represents the net horizontal force per unit area due to the slope of the free water surface. R. is the time mean averaged shear stress which must be X included in any realistic treatment of the surf zone where dissipative effects occur. Considering the shear stresses on a column of water (see Figure 4.1) and integrating over the depth, R^^ can be expressed R. = n n dz + dz i,j = 1,2 (4.20) d Z where t.. includes the combined stresses of waves and cur- rents. The shear stress on the horizontal plane can be integrated over depth, such that Figure 4.1. Shear Stresses Acting on the Faces of a Water Column 84 85 3t dz T .- T, . (4.21) where the subscripts refer to surface and bottom. The Leibnitz rule of integration must be employed to integrate over depth the stresses occurring on the vertical faces of the water column. Evaluating the shear stress on the vertical faces and taking the time average of the terms 3t 3 X il d: 8x, an 9h ^ji^^ " "^jin ~ ^jih 3x. -h -h Thus, the general shear stress term is given by (4.22) R 3x T . . dz - T . . 31 3n - T 3h . . ^ . . ^ + T . T, . (4.23) 3in 3x^ jxh 3x^ ni — hi -h 3 " 3 i,3 = 1,2 The equation for the conservation of total energy can be partitioned into the energy contributions from the mean and fluctuating parts with the aid of the conservation of mass and momentum equations. In the following development, it is convenient to work with the energy balance' 'for the' fluctu- ating motion alone, which can be stated as per Phillips (94) . 86 ■ Mf . U-M? 3U. In separating the mean and fluctuating contributions , the mean energy density has been represented as the total energy density of an "equivalent" uniform flow with the same depth and mass flux as the actual flow, plus the energy density of the fluctuating motion, minus an additional term represent- ing the difference between -i^ie- BTiergy density of the actual mean motion and that of the equivalent uniform flow. Then, the first term of Equation 4.24 is the rate of change of the energy density of the fluctuating motion minus the correc- tion term. The second term represents the convection of the fluctuating motion F, minus the correction term. The trans- port of energy by the fluctuating motion is given by F. = P n 2 ul_ (l/2(u!^ + w'^) + g(z - n) + p) dz (4.25) -h which represents the rate of work done by fluctuating water particle motion (for turbulence, this would be the work done by the Reynolds stresses) throughout the interior region of the flow, plus the work done by the pressure and gravity forces. The last term on the left of Equation 4.24 repre- sents the rate of work of the fluctuating motion against the mean rate of shear. Dissipative effects of turbulence have 87 also been allowed where £ jLs the ra-te of energy dissipation per unit area. The energy budget for the mean motion is given by 1^ (1/2 U.M. + 1/2 pgCn - h^)) + 3S 3I- M. (1/2 G.2 . gW) . U 3^ = U.T^. . (4.26) The terms of this equation represent, respectively, the rate of change of the kinetic and potential energy of the mean motion, the transport of the total mean energy, and the rate of work by mean motion on the fluctuating motion and bottom shear stresses. In the development of the conservation equations, no restrictions were plac-ed on the wave slopes or amplitudes. Also, no restrictions were placed on the fluctuating motion so that the equations are equally applicable to wave or turbulent motion. 4 . 3 Review of Wave Field Equatiohs The descriptive equations for the wave field are de- rived by solving the linearized boundary value problem over a horizontal bed. This solution can then be extended to a higher order by perturbation techniques. The development presented here will retain terms to the second order in amplitude (first order in energy and momentum) and neglect all higher order terms. The wave solution can then be substituted directly into the conservation equations 88 providing a means .for describing the wave-induced mean motions. In making this substitution and dropping all terms of orders higher that the second, only knowledge of the first order (linear theory) wave water particle velocities and surface elevation is necessary. This is because, in expanding and then averaging over the period, the terms involving higher order quantities in velocity and surface elevation- go to zero. The pressure must be known to the second order in wave height; however, the average second order pressure component can be determined from the first order water particle velocities and surface elevation terms. Thus, only the linear wave solution small amplitude is required. In utilizing linear theory, it is assumed that the motion is irrotational and that the fluid is incompressible and inviscid (see for example Kinsman (96)). These have been shown to be good assumptions (96). The fact that the linear theory is a good approximation is demonstrated by its success in describing many observed phenomena (96). Assuming simple harmonic motion, the surface elevation is (95) n = Y (k^x^ - CTt) i = 1,2 where the wave-number components are k = k cosa X (4.27) k = k sina 89 The arbitrary angle of wave incidence a is measured between a line parallel to the contours and the wave crests. The velocity potential for the first order solution is given by (96) H cosh k(h + z) ^.^ _ (^_28) 2 ck cosh kh x x The velocity is related to the gradient of the velocity potential = - (4.29) H g i cosh k(h + z) ^^^/^ ^ qn^ U . = TT ^ — r — r-r cosCk.x. - at J (.M-.ou; X 2 c k cosh kh x x The frequency equation relates the frequency to the local wave-number and water depth (96) = gk tanh kh (4.31) The pressure can be determined as a function of the depth to the second order by integrating Euler ' s vertical momentum equation over the depth and retaining terms of second order. Only knowledge of the first order water particle velocities and surface elevation is required to determine the mean second order pressure term. The time-mean pressure is given as 2 — 1 2 tt2 sinh k(z + h) /,. p = - pgz - g-pa H 2 (4.32) sinh kh where the first term is the hydrostatic contribution, and the second term is the second order mean dynamical pressure component . 90 The group velocity .-of- ±±ie waves can be determined from Equation 4 . 9 (c ). = ^ = £ n ^ = c,n (4.33) g 1 k dk k k 1 where the transmission coefficient is n = 1 (1 + .^^^ ) (4.34) " 2 sinh 2kh^ For deep water (infinite depth), n = and, in shallow water, n approaches 1. The energy density of the waves is proportional to the wave height squared and is equally partitioned between the potential and kinetic energy, such that the total energy density is E = ipgH^ (4.35) Recalling that the group velocity expresses the speed of energy propagation, the energy flux of the wave motion is F. = E c . (4.36) Assuming negligible wave reflection and a mildly sloping bottom, Equations '.4.3 3 and .4. 3 5 "' can be substituted directly into Equation :4.35: to obtain the flux of wave energy. The mass transport of the waves can be determined by recalling Equation '4.15. in which the fluctuating velocity was integrated from the bottom to the surface. Since the water surface elevation is unknown, the integrand is ex- panded in a Taylor series about the mean water surface level n 91 M. = pul^ dz + -h 3ul (n) p(u!(n) + z -rr-^ + ...) dz (4.37) 1 o Z n where the time mean average of the mass transport below the mean surface is zero (first integral), and only terms of second order in amplitude will be retained. Substituting for the wave profile and water particle velocity of the wave motion, Equations 4.2, and ■:4.30:, the ftiass transport of the waves is TII = 5. _i (4.38 ) M . = 0 nu . — - — T- 1 2-r\ c k Simple harmonic motion has been selected to describe the waves because experience has shown that this solution gives fairly good results for the deep water case. Inside the surf zone, however, the approximation is not good, but the assumption of simple harmonic wave trains will be retained with the exception of certain modifications on the celerity and amplitude. The wave field could be specified using the cnoidal or other higher order wave theories, but the accu- racy gained would not justify the increased complexity of the resulting equations (93). To simplify the working equations a determination of mass transport and energy is first arrived at. The schema- tic of the coordinate system used is shown in Figure 4.2. 32 ELEVATION VIEW Figure 4.2. Schematic of the Surf Zone 93 Although a longshore current distribution is shown, no prior knowledge of its actual shape has been assumed at this time. Due to the absence of any y-dependence , the mass con- servation Equation (M-.H; reduces to = 0 (4.39) 3x Integration gives M = constant =0 (4.40) X The constant of integration is equal to zero since the beach forms a boundary in the x-direction. This then says using equation 4.17 that U = - = - cosa (4.41) pD pcD which states that there is a mean reverse current balancing the mass transport onshore due to the wave motion. This must be true everywhere, both inside and outside the surf zone, to ensure that there is no accumulation of mass or growth of currents in the y-direction in order to maintain steady- state conditions in accordance with the original assumptions. The determination of the energy distribution is neces- sary as a means of relating the wave heights at various depths. The energy, in turn, must be related to the local angle of wave incidence. The angle of wave incidence is affected by refraction which can occur since the waves are allowed to approach at an arbitrary angle of incidence to the bottom contours. The problem is further complicated 9^ .since a shear flow is allowed which can also produce wave refraction. In the present model this shear flow effect on wave refraction will be assumed to be a minor effect and hence neglected. This is comparable to assuming that the longshore directed current is small in comparison to the maximum wave orbital velocity. The general statement of the phase relation gives an expression r.elating. thje., frequency, mean current motion, and wave angle. Equation '. M-.?' can be expanded to give (D = cr + Vk sina + Uk cosa (h.^■2) The wave-number was shown to be irrotational 3k Sk ^ - ^ = 0 (4.43) 3x 3y Since the wave length and amplitude of the waves are indepen- dent of the y-direction, the gradient of the local wave- number in the y-direction is zero. 3k 3k Integrating the gradient of k^ in the x-direction gives k sina = constant = k^sina^ (4.45) o o where the subscript "o" refers to conditions in deep water. This is a statement that, for straight and parallel contours, the projection of the wave-number on the beach is a constant. Assuming zero current in deep water, a general expression for the celerity of the waves can be derived where, as before ; 95 0) = + V sina + U casa) (4.46) and utilizing Equation '!4.45? c = (—JS. V) sina - U cosa (4.47) sma o For U = V = 0, this expression simplifies to Snell's law for wave refraction. The conservation of energy for the fluctuating motion, Equation ;4.24'', can b-e expanded for steady-state conditions to give . U M ^ U M ^ f-(EU + Ec - -4-4 ^-4r-) + S 1^ + S ^ = - z (4.48) 3x X gx 2pD 2pD xy 3x xx 3x where the gradients in the y-direction are zero. This equation states that the change in energy flux, due to currents and waves plus the work done by the excess momentum flux on the straining motion, is equal to the energy dissi- pated by turbulence and work done on the bottom. The prod- uct of a stress times rate of strain is a quantity that can be associated with power per unit volume. The last two terms on the left of the energy equation can be interpreted in this context where the excess momentum flux tensor then represents a stress and the velocity-shear a rate of strain. Longuet-Higgins and Stewart (91) named the excess momentum flux tensor a " radiation stress . " The excess momentiim flux tensor can be determined by substituting the wave expressions into Equation '4.18' . In 96 terms of energy, group velocity, and wave speed, an expres- sion applicable to both inside and outside the surf ^one is given by c 2c E-^os^a + ^i—^ - 1) c 2 c 5. — S. sin2a 2 c 1 -S. sin2a 2 c c c E sin^a + |(2-^ - 1) c 2 c (4.1+9) Effects of turbulence have not been included. Referring to Equation (4.41) and (4.49), it can be seen that U and S are of order E. The first and last terms of Equation (4.48) involving the product of these terms are of order (E ) and, hence, will be ignored in this analysis. From the result of the conservation of mass, Equation (4.41), U = 0, so that the third and fourth terms of the energy X ' equation are zero. Substituting for S and F , and retain- ing only terms of first order in energy, the energy equation reduces to (4.50) 8 N J. -n sin2a 3V _ -r— (Ecn cosa) + En — ^ ir— - - e dX / dX where the substitution, c = cn, has been made. Substituting for the wave celerity, as given by Equation (4.47), yields (— - V) sina cosa - U cos a sma o 3V + En cosa sina = - £ (4.51) d X 97 Again recalling that U is .of. ordBr E, and that all higher order terms involving the product of U can be neglected in this analysis, the energy equation can be written °_ En( . ° - V) sina cosa + En cosa sina = - £ (4.52) 3x sm o Expanding and cancelling terms, gives V) I- En sina cosa = - £ (4.53) sma 8x o Far outside the surf zone, energy dissipation due to bottom effects can be ignored, and the energy losses due to turbu- lence can be assumed negligible. Since the term in the brackets of Equation ..;4.5'3' is nonzero, the result of the energy equation outside the surf zone, assuming no energy dissipation, is En sin2a = constant (1.54) The relative amplification of the wave energy is then given by _ n sin2a ^ = ° . ^ • (4.55) E n sin2a o Since the term involving U in the celerity equation resulted in the product of higher order terms that are not included. Equation '4.47' can be written with the additional assump- tion that the velocity V is minor outside the surf zone, c c = (~ — ) sina (4.56) sma o or in terms of the wave angle 98 c (4.57) sxna = sma c o o which is a statement of Snell's Law of Refraction (96). Equations (4.53; and '4.55:'give a complete description of the wave amplitude and direction outside the surf zone. Equation :4.4-7- can be combined with Equation '■4.55: to give This states that the onshore component of energy flux is constant outside the breaking zone. The changes in energy density E, or wave height, can be determined from this equation as a function of the local wave-number and water depth. For decreasing depth as the waves approach the shore, the local wave length and the angle of incidence decrease. The effect of shoaling is determined by the group velocity. The group velocity initially increases slightly so the energy density decreases; the group velocity then decreases resulting in a continual increase in the energy density towards shore. A maximum wave height occurs at breaking. Due to the change in wave angle, which is the result of refraction, the wave crests become more nearly parallel to the beach. The energy density is less for waves approaching at an angle to a constant sloping beach than for waves whose orthogonals are normal to the beach because wave refraction results in divergence of wave energy. E c cosa g c - E — cosa •= const o 2 o (4.58) 99 As has been shown, there is an excess of momentum flux due to the presence of the waves. For conservation of momentum flux, there must be a force exerted in the opposite direction such as a hydrostatic pressure force or bottom shear stress to balance this excess momentum flux. It has been shown that outside the surf zone the component of excess momentum flux directed perpendicular to the contours is balanced by mean water level set-down, and inside the surf zone that such a balance leads to a wave set up (97). These effects have shown (9 7) to have a minor effect (second order) on water depth outside the surf zone and for all practical purposes can be neglected in that region. The seaward edge of the surf zone is usually de- lineated by the point where the waves first start to break which is reached when the particle velocity at the wave crest is equal to the wave celerity. Inside the surf zone, the waves are unstable, and the fluid motion tends to lose its ordered character. Waves break in different ways, depending primarily upon the wave steepness and slope of the beach. The manner in which they break has a very definite influence on the hydrodynamics inside the surf zone which, in turn, affects such quantities as the sedi- ment transport, longshore currents, and wave runup. 100 Based on observations by Galvin (76), breaking waves are usually classified as spilling, plunging, or surging. Spilling occurs when the wave crests become unstable, curl over slightly at the top, creating a foamy advancing face. Plunging occurs on steeper beaches when the wave becomes very asymmetric; the crest curls over, falling forward of the face resulting in the creation of considerable turbu- lence, after which a bore-like. wave. front develops. Surging occurs when the wave crest remains unbroken while the base of the front face of the wave, with minor turbulence gen- erated, advances up the beach. There is a continuous gradation in the type of break- ing, and Galvin (76) found it convenient to add a fourth category of collapsing to describe a type intermediate to plunging and surging. He performed extensive laboratory investigations to quantitatively classify breaking waves according to the wave and beach characteristics . Combining his results with earlier works, he grouped the breaker type depending on beach slope tanB, wave period T, and either deep water or breaker height H. The breaker type can be established as shown by Battjes (77), by the dimensionless (H /L )-^2 (H^/L^)^^^ variable, — 2 — 2 or , . As either of these tanp tanp parameters increases, breaker type changes from surging to collapsing to plunging to spilling. Spilling breakers are associated with steep, relatively short period waves and 101 flat beaches; plunging breakers ..are .a^-sociated with waves of intermediate steepness and the steeper beaches; and surging breakers are associated with waves of small height and steep beaches. On natural beaches, breakers classed as spilling are most commonly observed, followed in decreasing order of frequency by plunging, collapsing, and surging. In the laboratory tes-ts, spilling ..bjr>eakers are relatively rare compared with collapsing and surging breakers because slopes used in laboratory tests are usually steeper than slopes commonly found in nature due to the physical limita- tions of space. The breaking index curve provides a relationship be- tween the breaking depth , the breaking wave height , and the wave period T. In shallow water, the relationship simplifies to H'^^/D^ = < , a constant. Longuet-Higgins (M-6) compiled breaking wave data from several sources (9 8-106) including lab and field data. He found a fairly good correlation for breaking waves as compared to the breaking wave criteria predicted by the solitary wave theory. The solitary wave theory predicts a value for k of 0.78, while other theories predict slightly different values. Theo- retical values range from 0.73, found by Laitone (105), fo cnoidal wave theory, to a value of 1.0, found by Dean (10 5 using a numerical stream function theory. Experiments on steeper laboratory beaches show that the value of k can be 102 much larger. All th.e theor-etical values have been calcu- lated, assuming a flat bottom, and correspond to beaches with very gentle slopes. A summary of various investiga- tor's findings zis presented in Table 4.1. The important point is that the breaker height is governed by the depth of water . The energy loss due to the wave breaking process is dissipated in the generation .ol turbulence and heat, bottom friction, dissipation, percolation, and viscosity. The waves in the surf zone constitute a non-conservative system in which the use of potential flow theory is no longer valid. In fact, there is no analytical description avail- able for the waves in the surf zone. Gross assumptions are required. The linear wave theory will be retained as the input to the conservation equations, but with modification to the wave amplitude. . The wave height inside the surf zone as noted is controlled by the depth of water and is of the same order of magnitude. Spilling breakers lend themselves to a physical treat- ment since the potential energy and momentum flux of the « waves inside the surf zone can be expressed approximately in analytical form. If the beach slope is very gentle, the spilling breakers lose energy gradually, and the height of the breaking wave approximately follows the breaking index given previously. The height of the wave is then a function of the depth. 103 H, /2 Table "+.1. Observed and Theoreirical "Values of j = . Investigator tanB K 2 IC 2" Observed Values Average Putnam et al. (44) 0.065 0.098 0.37 0.36 0.139 0.143 0.32 0.37 0.35 0.144 0.32 0.241 0.35 U . Z D VJ Iverson (98) 0.020 0.41 0.033 0.050 0.38 0.42 0.44 0.100 0.52 Larras (99) 0.010 0.34 0.020 0.091 0.37 0.43 0.39 Ippen and Kulin (100) 0.023 0.60 0.60 Eagleson (101) 0.067 0.56 0 . 56 Galvin and Eagleson ( 54 ) 0,104 0.59 0.59 Bowen (47) 0.082 0.45-0.62 0 . 56 Values Determined from Solitary Wave Theory McCowan (102) 0.000 0.39 Davies (103) 0.000 0.41 Long (104) 0.000 0.406 Values Determined by Cnoidal Wave Theory Laitone (105) 0.000 0.37 Values Determined by Numerical Wave Theory Dean (106) 0.000 0.50 104 In the present model, it will be assumed that the waves act as spilling breakers inside the surf zone and that they follow the breaking index, k = 0.80. The wave height inside the surf zone is then given by H = kD (4.59) It is further assumed that the kinetic and potential energy are equally partitioned so that the total wave energy can be described in .texims of the wave height which is a function of the depth E = I pgK^D^ (4.60) This is a non-conservative statement of the energy distribu- tion within the surf zone. The waves inside the surf zone are assumed to retain their simple harmonic character so that the wave profile and water particle velocity are described by n = Y cos (k^X^ - (jOt) (4.61) u = 75- ^ r-^ cos (k.x. -(jot) 2 c k 1 i The expression for the horizontal water particle velocity is based on the Airy wave theory and has been simplified for shallow water. In very shallow water, the waves are non-dispersive with the wave speed being only a function of the depth. In keeping with the linear theory approach to calculation of wave energy it will be assumed that wave celerity is ade- quately described by linear wave theory 105 c = (gD)-^^^ (4.62) It has been found experimentally in shallow water breaking, wave celerity can be better approximated by soli- tary wave theory (102); c = (g(H + D))^''^ = (g(l + <)D)^^^ (4.63) It is felt though, that since wave height is linear in depth any way and therefore there is only a difference of a con- stant between the two descriptions of celerity, the linear wave model should be used for consistency of theory. M- . M- Proposed Sand Transport Model 4.4.1, Longshore Current Estimation The derivation of the longshore current to use in the sand transport equation parallels an approach similar to Longuet-Higgins (46), and Thornton (30), in that a linear- ized bottom stress in longshore velocity is found from which longshore current can be postulated. The present approach is different only in that a wave orbital velocity shear stress friction factor is used in the bottom shear stress term, and wave set-up is included. In order to formulate the longshore current problem some assumptions concerning the flow conditions are restated. In the present development as in the simplified models of Longuet-Higgins (46) and Bowen C47) it is assumed that (1) longshore currents are steady, no time dependence, 108 (2) homogeneous and incompressible fluid; (3) hydrostatic pressure -.CLO 7>, l4)rCD'rioli-S force is negligible; and (5) currents are sufficiently small such that wave current interaction is negligible. Using the assumption of uniform motion in the y direc- tion and no time dependence. Equation 4.16 can be reduced to the following forms 0 = - pg (n+h)4^ - T, + T ' 8x hx rx (4.64) n -h where as T . - T-, + T xydz hy ry "rx 3x SS (4.65) T 'ry 3x and D = n + h (4. 65) The quantity under the integral sign can be posed similar ily to a turbulent Reynolds stress as T dz = y D 1^ (4.67) xy ex dx -h Then assuming that the onshore time average bottom stress is zero, equation 4.6 4 can be reduced to 107 0 = -pg(n+.h'>-|^-+ T rx (4.68) 8x ex dx hy ry From equation 4.4 9 with the shallow water assumption it can be found that S^^ = I (3 - 2sin^a) (4.69) and c S = E cosa sina (4.70) xy c Also, a relationship for Snell ' s Law of Refraction (i.e., Kinsman (96)) is of the form sina sma o = constant = (4.71) c c o which can be included in the above term to obtain sina S = F (4.72) xy X c , o where F is the onshore directed component of wave energy flux F = E c cos a (4.73) X g and was shown to be constant outside the wave breaking zone. Therefore, the quantity S is also constant outside the xy wave breaking zone and eaual to the deep water value S , xy where ^xy ~ ~2" ^°^^o ^-^•'^'^o (4.74) 108 Figure 4.3. Schematic Plan and Section of the Nearshore Region 109 At the shoreline (where D=0) since wave height H is proportional to depth D the above quantities become F = 0 X S =0 xy and the total external lateral force on the breaking zone equals E Total lateral forc^. on .breaking zone = — sin2a^ (4.75) Considering now the balance of momentum in the onshore direction of the water between the breaker line and the shoreline as expressed by Equation 4.6 8 with the expression given previously H = kD = ic(n + h) (4.76) one can derive the following equation dx 2(3-2 sin a, ) , K b d 16(n + h) dx (n + h) (4.77) or = _ K = - K tang . dx dx where K 2 2 K (3 - 2 sina, ) b 8 + ic'^O - 2 sin'^a, ) b (4.78) (4.79 ) If a new horizontal .-cGordinate , x, is introduced, with its origin at the point of maximum set-up x = ~^s' "then X = X + X , and rf + h = (1 - K) tan 6 x. s Assuming "=0 at the breaker line, one can integrate equation 4.77 to find n" and the total depth D across the 110 surf zone. In reality, it has .been.-shown that there is a setdown at the breaker zone (i.e., rf = negative value) but the set down has been shown to be negligible for all prac- tical purposes (97). Inside the breaker zone F gradually diminishes toward the shoreline. A consideration of the momentum balance between two planes x = x-^^ and x^ + dx parallel to the shore- line and - separat-ed by a distance dx shows that the net stress T^^ per unit area exerted by the waves on the water in the surf zone is given by ^ry = - ^S^y/^^ ^^'SO^ and using equation 4.72 this equation becomes 8F sina O 3^x where g— ^ denotes the local rate of energy dissipation in the onshore direction. This equation notes that the local stress exerted by the waves is directly proportional to the local rate of dissipation of onshore directed wave energy. Outside the breaker zone the mean stress vanishes as S is xy constant as shown previously. In some situations the loss of wave energy can be attributed to bottom friction (due mainly to the orbital velocity of the waves). Observations by Munk (108) that in the surf zone the breaker height is proportional to the mean depth suggests that under normal circumstances most of the Ill loss of wave energy is.due.io wave breaking though not to bottom friction. It is important to remember that the large portion of the wave energy expended in the surf zone is not used in moving sand in the longshore direction but is dissipated mostly through the turbulence mechanism. Other investi- gators have correlated the entire energy flux dissipated throughout the- ^surf ^one- ■■with the amount of sand transport moved as suspended and bed load (30), while only a small fraction of this energy is readily available for actual sand transport in the longshore direction. It was shown previously that H = kD where k is a constant between 0.7 and 1.0 (see Table 4.1). Using linear wave theory where kD<<l we have c = /gD - c and therefore we have T- tH^ 3/2 ^5/2 ,„ F = ~— c cosa = -5— pg D cosa (4.82) X 8 g 8 '^'^ and from equation 4.81 3 S rn xy 5 2 / T^^3/2 dD sina cosa T = - ^ ^ = - K p ( gD ) -3 ry 3x 16 * dx c = - <^yD 4^ sina cosa (4.83) 16 ' dx where ^ = ^ = (1-K) tan3 is the modified local bottom slope, dx 112 Using equations H.82 and the linear shallow-water theory, we can also express equation 4.83 in terms of the maximum horizontal bottom orbital velocity given by ^hwm = f '^/^^ = f^"/^^ = f ^ (4.84) Then we have simply 5 2 = - IT P^hwm ^^^^ sina cosa (4.85) Seaward of the .breaker line the onshore component of energy flux is constant (i.e., no dissipation assumed in the present case) and thus T =0 ry The tangential stress excited by the water on the bottom will be assumed to be given adequately by a relation of the form = f f |u^ |u^ (4.86) Where u^^ is the instantaneous velocity vector near the bottom and f ' is a friction factor. If there were no longshore velocity, and if the ampli- tude of the motion were small and the bottom impermeable, the horizontal orbital velocity would be expected to be to- and-fro in the same straight line, making an angle with the normal to the shoreline (see Figure 2a in Longuet-Higgins (46) ) . The frictional stress would then be given by "^h = f l^hw l^hw (4-87) 113 which would then vanish ±n ■ the, mean - BJZcoTdlng to linear wave theory. Here f , has been assumed for f since the motion is w entirely due to waves . If a small component of velocity in the longshore direction V is added to the orbital velocity, the frictional stress no longer vanishes in the mean. Assuming a is small, the component of velocity V is almost perpendicular to the orbital velociiiy. To first urdeT, the magnitude of the velocity l^hl = ^l^hwi' ' \^\^^^^^ (4.88) is unchanged but the direction of the bottom stress is changed by a small angle - V/|u^^|. This leads to an addi- tional stress in the y direction given by V = J l"hwl l%„l 1:^1 ' ("-sa) Physically, when the orbital velocity is onshore, the direction of the bottom stress is inclined more tovzard the positive y direction (when V is positive); when the orbital velocity is offshore, the bottom stress, now almost in the opposite directions, is again more toward the positive y direction. This is shown in Figure 4.4. Taking the time mean value of the longshore shear stress the following relationship is found: 114 V (4.90) hy 2 as is assumed sinusoidal as per linear wave theory TT ^hwm (4.91) As the friction coefficient was determined in accordance with a small V comparatively to the orbital wave velocity u^^, the bottom shear is dominated by the wave ortibal shear stress and the proper friction coefficient is assumed to be f ^ (i.e., the bottom shear stress is governed by the wave w ^ ^ orbital motion being the dominant shear stress term) . The friction factor f has been found (65) as a func- w tion of the bottom wave particle amplitude Reynolds number and the roughness to wave particle amplitude ratio. The equation of motion in the longshore direction can be written as before 8 . ^ 9V. 0 = T , + ^ (y B--^) - T, (4.92) ry 9x ex 3x hy where in the surf zone T^^ and x^^ are given by equations 4.85 and 4.90, respectively. Little is known as to the lateral "mixing" coefficient U which spreads momentum laterally across the surf zone. A few theories exist, but are based on very limited data (109). As the spread of lateral momentum across the surf zone only changes the lateral distribution of the longshore velocity and its maximum value, and the required final 115 Figure 4.4. Shear Forces Due to Oscillatory Wave Velocity and Longshore Current 116 velocity distribution is to ha J.ntagi?ated -across the surf zone, it will first be assumed that y is equal to zero. This is equivalent to saying that there is no shear coupling between adjacent water columns taken across the surf zone. This may be contrary to what various experiments on long- shore currents have shown (109) but will be sufficient for initial development of the intended longshore current equa- tion. It will l-airer be -shown that inclusion of an u value not equal to zero and proper accountability for the longshore current distribution across the surf zone only changes the integrated sand transport equation by a constant, Another way of neglecting the lateral eddy viscosity in the longshore current equation is to suppose that the ex- change of momentum by turbulence is negligible in comparison with that due to waves (M-S). Then, in general the second term on the right of equation 14, 92 can be neglected in comparison with the first. There remains a balance between the first and third terms : T7~ = T (4.93) hy ry Substituting from equations 4.8 5 and 4.90, we have in the breaker zone -f£-u^ V = -^pu, ^ (1-K) tanS sina cosa (4.94) TT w 2 hwm 4 nwm and 117 V = ^-J- u, (1-K) tan-B- sin-a 'cosa (4.95) 4-1 nwm w Using the linear theory relationship between and wave celerity c the equation for longshore current velocity becomes V = ~ ^ gD(l-K) tang (iiBE) cosa (4.96) w where c = /gD as before. By Snell's law the next to last factor in the above equation is constant. Thus, in this derivation, the longshore velocity is simply proportional to local total depth D. If it is assumed that the shallow water theory is valid as far out as the breaker line where the depth D is equal to Dg , the longshore current, in the absence of horizontal mixing, can be written as ° < °B 0 ^ ^ (4.97) where Vg = velocity at breaking line = 1^ ^ /gU^ (1-K) tang sina^ cosa^ (4.98) w The velocity distribution represented by equations 4.9 7 and 4.98 is a triangular distribution with a maximum veloc- ity at the breaker line. As noted earlier the turbulent mixing as represented by a "mixing" eddy cofficient y has been arbitrarily assumed as zero which provides a 118 discontinuity in the velocity. distri±i.utioji .profile at the breaker zone. Both Bowen (47) and Longuet-Higgins (1+6) have solved for the case of lateral shear mixing across the surf zone by assuming an eddy viscosity based on mixing length theory. The present extended model exactly parallels the solution given by Longuet-Higgins (46), the only difference being that wav^ set up is account-ed for. In Longuet-Higgins solution (46) and the present solution y is assumed to be given by a form of mixing length times mixing velocity = Npx /gD (4.99) where N is a dimensionless constant; x is a mixing length, increasing linearily from the shoreline boundary similarly to the Prandtl hypothesis; and, the mixing velocity is proportional to the horizontal water particle velocity which is proportional to wave celerity = /gjD. The solution of equation 4.9 2 under the consideration of lateral shear mixing is given in (46) and the solutions are shown here with the modification that wave set-up is included. The equations for longshore current distribution with wave set- up are V„ (B.X ^1 + AX) 0< X<1 V = { ^ ^ p (4.100) Vg (B2X^2) 1< X < <» and by matching solutions at the breaker line 119 x=l V x=l 8x x=l 3 x = l 3x (4.101) the following values are found 1/2 . + L> 4 '16 p = . 3 , (4 , 1, „ _ 3 , 9 , 1.1/2 B- B, A . A and where P2-I P1-P2 Pl-1 P1-P2 2_ 2-Sp 2ttN(1-K) tanB KT (4.102) (4.103) (4.104) (4.105 ) (4,106 ) (4.107) w X = x/x, a dimensionles s distance x^ = distance to the breaker line from the shoreline As can be seen in the above equations at the value of p = 2/5 a singularity exists in the above solutions. The solution at p = 2/5 is of a separate form (46), and is given by 0<X<1 (4.108) l<X<oo V = Vg (10/49X - 5/7X In X) Vg (10/49X - 5/2) As before, Vg equals the longshore current velocity at the breaker zone in the absence of lateral shear mixing and is defined in equation 4.98. 120 Experimental evidence in laboratory longshore current experiments (46) suggests a coefficient of p between 0.1 and 0.4 when comparing laboratory longshore current measurements to theoretical values. 4.4.2 Integrated Sand Transport Estimation The proposed sand transport model is an extension of an existing model of sand transport based on an energy-rtype- approach due to Bagnold (45) and postulated for unidirec- tional current flow. The present model extends Bagnold 's basic concept to the realm of sand transport along beaches parallel to the shoreline due to wave action. The concept can be extended to the case of sand transport along beaches due to other driving forces such as wind shear on the surf zone or shear due to tidal induced currents (i.e., hydraulic tidal currents in the vicinity of inlets). In the presently developed model though ' only the sand transport due to wave motion will be considered. The presently developed model also differentiates between bed load and suspended load sand transport and each component can be obtained independently, a distinct advan- tage not realized in many sand transport models (i.e., Einstein (110)) where the suspended load component is dependent upon the calculated bed load component. Now, defining the dry mass of a unit area of material to be m and the solid density of the material to be p , the 121 immersed weight of the tmit . ar ea of ied material is m (— ^ — )g. This immersed weight of sediment can be divided Pg-P into two parts m^ and where the symbol m' = m(-^ ). The b.ed load m^ is that part of the load which is sup- ported wholly by a solid-transmitted stress m^g and the suspended load m' is that part which is supported by a fluid-transmitted stress m'g. The total transport rate for the immersed mass of material per unit width of bed is H = "bs ' ^ss h ^^-^^^^ where iT-j^ = average velocity of the sand particles in the bed load and U = average velocity of the sand particles in the suspended mode of motion i^ = bed load immersed weight transport rate per unit width bed i = suspended load immersed weight transport rate per ^ unit width bed The dynamic transport rates i^ and i^ have the dimensions and quantity of work rates, being the products of weight force per unit bed area times velocity. As these quantities stand, they are not in fact work rates though for the stress is not in the same direction as the velocity of its action. The dynamic transport rates become actual work rates when multiplied by conversion factors and A^ , each defined as a ratio 122 tractive stress needed ±q Tnainj-^-iTg transport of the load normal stress due to immersed weight of the load (4.110) For bed load the factor A^^ is a friction coefficient tan(}>, where tan(j) expresses the ratio between normal stresses on the bed and lateral stresses opposing motion parallel to the bed (which is some portion of the normal stress as in solid physics). The forces acting on a unit area of bed are as shown in Figure 4.5. The analogy in solid physics can be expressed with a solid body on a flat surface. The coefficient of solid friction at rest is most easily measured by resting one body upon the other and increasing the angle of inclination of the shear plane until shearing begins by gravity. Then since gravity force parallel to slope = mg sine}) , and gravity force perpendicular to slope = mg cos^ , the coeffi- s , parallel force ^ a j. j. ■ c cient — —f- — -r-. =. ^ = tan({) = static angle of repose perpendicular force s ^ ^ of sand. In solids the dynamic coefficient of friction is lower than the static coefficient of friction; therefore, the expressed angle cp would not be expected to be equivalent to the angle of repose except for the limiting static case which provides an angle of repose = 3 3° for which tan(j) = 0.63. s Bagnold (45) found that the dynamic friction coeffi- cient or stress ratio across shear planes of an array of solid grains is of the same order as the static coefficient 123 Lift = L Prog Weight=G, (G3-L5) tan<^ Gs-Ls =Dispersive Stress due to Normal Grain Momentum G.-U Friction Force Resisting Movement of Sand Grains / / / / = (Gs-L3)tan<^ CDrcg (for Sediment Movement) Figure 4.5. Relation of Normal Force to Moving Force 124 when the grains are closely packed and also when they are considerably dispersed. The actual value of tan(}) was found by Bagnold (4-5) to be a function of a parameter for dispersed solids in a fluid and is analogous to a fluid Reynolds number where ^s = f S U ^ and X is a linaar . spacial. concentration defined by the ratio X - mean diameter d of solids mean free distance between solids The bed' load work rate then becomes bed load work rate = m'g U, tan<J) = i, tan4> (4.111) b° bs b since the work rate must be the resistance force (in the direction of the moving grains) times the velocity of the moving grains . The suspended load work rate can be inferred more simply. The suspended solids are falling relative to the fluid at their mean fall velocitv w , but the center of s ' gravity of the suspension as a whole when time averaged does not fall relative to the bed. Thus the fluid must be lift- ing the solids at the velocity w . The rate of lifting work done by the shear turbulence of the fluid must be ^s suspended load work rate = m'gw = i, — — (4.112) S S D ^7 ss which points out that the factor A is equivalent to 125 w A = = U ss • When any kind of continuing work is being steadily done the principle of energy conservation can be expressed in terms of the time rates of energy input to, and output from, a specified system by the equation rate of doing work = available power - utilized power or in an equivalent alternative form rate of doing work = available power x efficiency (4.113) d Work dt = P For bed- load transport this equation becomes ±^ tan<|) = % ' '^■^ (U.im) or e, = :rf—r (4.115) b tantp b with e^^ an efficiency factor for bedload (as defined in Bagnold (45)) less than unity. As the bed load force used to determine bejilload work rate is in the direction of trans- port, the available power P^ for bed load transport in the longshore direction refers to the power expended on the bottom in the longshore direction. The power used to support the suspended load is e P s s therefore, the equation for the suspended load transport component becomes 126 w i — = e P ■ (4.116) s u s s ss or _ U ss T3 1 = e P 3 8 W S S The proper quantity of available power to use in this case P is the power available for suspension of the sand, not 5 the longshore power as in the bed load. More will be men- tioned concerning this power quantity later. Adding components of transport, the equation for bed load and suspended load transport combined becomes e. e U i = i, + i = (3^-^)P, + ( ^ ^^ )P (4.117) b s tanc}) b w^ s It will be assumed that the mean velocity of the suspended solids U equals the mean velocity of the fluid which for a steady longshore current becomes V. It remains to discuss the available power in terms of wave motion and the efficiency factors for the various transport modes. Available power for transporting sand in the longshore direction for bedlload is that power dissipated by the bottom stress in the longshore direction. In the analogous case of riverine sand transport the power dissipated would be equal to Available stream power = x, U 127 where x^^ is the bottom stress per unit area of bed and U is the average velocity of the stream. For the case of wave motion in the surf zone the power P, becomes D Available longshore power for bedload transport = t^^ • v = (x) (4.118) where x, = the bottom stress exerted by the combined wave and c-urrent action ±n the longshore direction V = the longshore velocity In the case of no lateral mixing the longshore wave induced stress would be from equation 4.93. 3 S T - _ S 2 ^ _,>,3/2 T,v, „ sina cosa , n \ hy " 3x 16" P^g^-* (1-K)tan3 (4.119) and as shown previously in equation 4.9 6 V = ~ J- /gD^ (1-K) tanS sina^ cosa^ ~- (4.120) w B Using equations 4.118, 4.119, and 4.12 0 and invoking the shallow water assumption and Snell's Law, the total long- shore power available for bed load transport becomes P^(x) = |i(i)p|- . (gD)^^^ • (1-K)^tan^g( ^ ^ )^cQsa^ cosa w b (4.121) For the case of suspended load sand transport the analogy between the available power in riverine system and the surf zone no: longer holds . For the riverine system the 128 available power for suspension of sand is the same as the available power for bed load motion. For the surf zone though, complex processes act to entrain sediment into the flow system such as breaking waves plunging on the beach and creation of turbulence by surging wave action ( similar to "bores" in a river). In this case the available power for suspension is the total power expended by the waves in the surf zone. .Th,us-y. at- a,, spexiific .location in the surf zone the available power for suspension of sand is P = I- (Ec cosa) (4.122) s dx g where Ec represents the energy flux per foot of wave crest and the cosa factor resolves the power per foot of beach rather than per foot of wave crest. The quantity P then represents the onshore gradient of the onshore energy flux. The total sand transport due to bed load and suspended load is, from equation M-.115 with U = V s s e, e V i = i, + i = (t^^) P, + (— ) P^ (4.123) I h s tan(J) b w s s where now ^hy (x) 25tt 3 IC 16 8f w V(x) s inct (gD)^'^^ (1-K)^ tan^S (— — -)'^ cosa^ cosa b (4.124) and P (x) = I- (Ec cosa) = T-|<^T ^ D^^^ (1-K) tang cosa(4.125) S dX g 16 129 Upon integration of these quantities across the surf zone the total transport becomes (assuming the efficiency factors are constant across the surf zone) ^t) s e I. = / (i^ + ie) = ^ir|-r) P>,(x)dx+-^ VP^(x)dx X o D s tanqj o d , w o s /•■I o s-K sin2a, e^c, cosa, = (K <(1-K)tang ^( b b ^ _s_b ^).p^ (1+.126) w f tan*' 2 w I w ^ s where yH, 2 sin 2a, and K = constant = 1.40 w which can be rewritten as where /-n o s, sin2a, e„ c, cosa, = (<(1-K)tane) . (_b b ^ _s_^ b) (4.127b) ^ r tan4) • 2 w w ■ s At this point it is interesting to compare equation 4.126 with the "modified" presently used sand transport relationship of Reference (25) discussed earlier, where yH, 2 sin 2a, ■n b b ^5, 8 ^b 2 Equations 2.1 and 4.12 7 can be equated by the relation- ship 130 tanip 2 w s CK, — ^ tan3) w X W K X (1+.128) Using "rough" values of efficiency factors with e^^ = 0.13 from an average of Bagnold's (45) range of values for bed load efficiency in rivers and e = 0.002 "postulated" in a heuristic model for suspended sediment transport by waves (Dean (26)), and with the following assumed values for typical wave and sediment parameters tan(|) = 0. 63 = 0. 20 ft. /sec. (d=0 ^b = 2 feet ^b ~- 8° f - w 0. 02 tan3 = 0 . 01 a value of ^Ib can be found ^Ib" 0. 50 As noted in earlier, the sand transport computations is 'ib ~ ^'^ although many inves- tigators (111) believe that the present value used is too high and should be on the order of 0.M--0.6 based on reanalysis of past data. It therefore appears that the presently postulated sand transport formula appears to provide a reasonable approxi- mation for the prediction of sand transport on beaches. This will be checked out further later. 131 4. '1. 3 Sand Transport Distribution Across Surf Zone Now, using the previously developed (4-6) distribution of longshore current velocity as given in equations 4.100 and 4.10 8, and the presently developed sand transport relationship given in equations 4.12 3, 4.124, and 4.12 5, the distribution of sand transport in the lateral direction (across the surf zone) can be calculated. . .Again the equation for' sand transport can be written i . i + i = ^ . (T V) + !il 8(EPg cosa) X, b s tan<p hy w^ 3x where from equation 4.9 0 ~-^y t)^ written T,„ = f , (J /pT /X) V (4.129) hy w 2 TT ° D where X is — , a dimensionless distance from the shoreline ^b and °b ^b ^b Tl-K) tang ^ <(1-K) tanB (4.130) This equation can be written in terms of the dimensionless variable X as , _ ^b ^ ^^w'^b „l/2„2 P — ^ ' I tan^ ^ 2¥ + — (y|- c, D, (1-K) tanBcosa) X^^'^V (4.131) w 15 b b s Now the above mentioned longshore current model (which incorporates lateral mixing) as discussed in equations 4.100 and 4.108 can be used in the above expression. It should be 132 noted that outside the surf zone there can still be bottom friction in the case of lateral shear mixing, as the bottom friction balances the gradient in lateral shear force be- tween adjacent water columns (since the gradient of the "radiation" stress is zero). Hence, bed load can exist outside the surf zone. As the available power to suspend the bottom sediment is considered to be due to the wave breaking process though, there Is no suspended sediment transport outside the breaking zone. Using equations 4.98, 4.12 3, 4.124, and 4.12 5, the equation for sand transport as a function of the distance across the surf zone can be written as (after considerable manipulation and use of the shallow water assumption, along V with a dimensionless longshore velocity defined as V = tj— ) B e, sin2a, i m ^% Han(i)-2 e^c^cosa -^3/2^ ^25tt^ ^ K:(l-K)tanP ^ ^ (4.132) w 16 f x, s w b which can be written in dimensionless form as 777TT£K7tanF7 ^16 ' ^ tan(j)-2 f ^ w e c, cosa, a. ^ / o ^ V X^'^^} (4.133) w s The distribution of sand transport across the surf zone can now be found using assumed values of the efficiency 133 factors and other appropriate wave and sediment parameters. As an example the following values are considered: e, = 0.13 D tancj) = 0.63 W : S ■ 0. 20 ft. /sec. e : s : 0. 002 : 2 ft. '- : 8° : 0. 2 A dimensionless plot of the sediment transport lateral distribution for the above assumed values is shown in Figure 1+.6. As referred to previously, there is no allowance for suspended sediment transport outside the surf zone in the present model since the energy dissipation (available power for sand suspension) is zero outside the region of breaking. This leads to discontinuity in the sand transport distribu- tion at the surf zone. The question as to whether such a distribution occurs in reality cannot be answered as no data existjwith which to compare such distributions although at least one researcher, Ingle (75), found that little sand transport occurred outside the surf zone during flourescent sand tracing studies. Other researchers (73, 7 4) make reference to the absence of sand transport seaward of the wave breaking area based on laboratory results although no quantitative data aire given . From color aerial photography 134 of the surf zone it also appears that the breaking wave zone provides a clear demarcation between the "colored" water suspended load area inside the wave breaking zone and the "clear" water area outside the wave breaking zone which is at least qualitatively in agreement with the present model (i.e. lateral mixing of sediment seaward of surf zone is negligable). Using the postulated sand transport lateral distribu- tion, the sand transport can be integrated across the surf zone (assuming a mixing coefficient value) and a "refined" value of coefficient can be found. Again using equation 4.131 and integrating across the surf zone a dimensionless equation is found; , 25tt , K(l-K)tan6^ 16 ^ •) e, sin2a, , tancp • 2 w (4.134) e c,cosa, , w which can be rewritten as I ^ ic(l-K)tang ^ f ^ w (^) ^16 ^ e, sin2a, .,. ^ e c, cosa, j' (-h ^)(i" + t") + ( s ^ ^ ) i" ^tant})-2 ^^^i^^2^ ^ w ^ ^3 (4.135) where 135 136 I* = /^V^X^'^^dX (4.136) o I* = TV^X^/^^X . (4.137) ^ 1 I* = /^VX^'^^dX (4.138) o using the solutions for V presented in equations 4.100, these integrals can be integrated to provide the following values for "p" arbitrary (but i 2/5) * B^^ 2AB^ 2A^ ^1 = (ip^+3/2) ^ (p^+5/2) ^ T" (4.139) ft -"^2 ^2 ^ 2p2+2 (4.140) for "p" = 2/5 ij = 0.0594 I2 = 0.0139 (4.142) I* = 0.1166 The values of these integrals have been tabulated and are presented in Table 4.2. 137 Table 4-. 2. Integrals for Sediment Transport. A A p V^2 T ^3 • UU± OC A A A A O C A C A C . 2635 A T A C . 2706 • Quo A 1 1 A a . 24-28 A A C O .0058 A P 1 O . 2486 . 2603 A 1 O T . 2137 A A A A . 0092 A O O A . 2229 A 1 1 T A . 2419 .1543 AT 1 ) 1 1 . 014-4- . 1587 A A A C . 2005 -10 T A A O . 1228 . 0158 n o A c . 1386 .1758 . 15 . 1037 .0160 .1197 . 1596 . 20 A A A A . 0902 A T C A . 0158 1 A A A . 1050 .1474 . 25 A T n n . 0799 . 0154- . 0953 .1377 . 30 . 0718 . 014-9 A A . 0867 .1296 .35 .0651 . 014-4- . 0795 .1225 . 40 .0596 . 0139 . 0735 .1167 . 45 .054-7 . 0134- . 0681 .1113 . 50 .0505 . 0129 . 0635 .1055 . 55 A 1 1 T A . 04-70 . 0124- A C A 1 1 , 0594 T AAA . 1022 . 50 . 04-09 . 0115 . 0524 . 0948 . 70 . 0384- . 0110 . 0494 A A T C . 0915 . 75 . 0361 AT A . 0105 A 1 r T . 0457 A A O C . 0885 . oO A A 1 1 A . 0340 A T A A . 0103 . 0443 A O C "7 . 085 / . 85 . 0322 . 0099 . 0421 . 0831 .90 .0304- .0095 .0399 .0806 .95 .0289 .0092 .0381 .0783 1.00 .0274 .0089 .0363 .0752 1.20 .0227 .0078 .0305 .0687 1.50 .0175 .0064 .0240 .0601 3.00 .0071 .0031 .102 .0373 6.00 .0024 .0012 .0035 .0214 10.00 .0010 .0005 .0015 .0137 138 Estimation of .£f f i ciBiicy Factors and Dynamic Friction Angle It remains to discuss appropriate values of the effi- ciency factors e^ and e^ for the case of sand transport and the dynamic coefficient of friction for bed load movement. The suspended sediment efficiency factor e^ has been defined previously as work rate necessary to maintain a suspended load s available wave power per foot of beach (14.143) S 3x —5- • c • cosa) 0 g = Ss-1 where m' of equation 4.109 = C m^ (— f — ) s ^ f Sg and C = time and spacially averaged (over depth) dry weight concentration of sediment mjT = mass of fluid in a unit area of surf zone at a given depth = pD Thus e^ can be reduced to the following equation e = ^ s (4.iLm) S o 2 c cosa(l-K)tanS 16 g To determine a value of the factor e^ the problem becomes one of postulating a method for determining the time and spacially averaged concentration over depth using wave and sediment parameters. Heuristic models will first be pre- sented to define the important parameters and then a 139 semi-empirical approach based on existing observations of suspended load in waves will be used to formulate a model for finding the concentration. Assume first that a breaking wave crest suspends mater- ial to a maximum height in the water column which will be considered proportional to the breaking wave height = 6H as shown in Figure Now, if the time required for the grain to fall back to the- boirtroTii -t. = — is less than the b w s wave period T, then the depth averaged concentration as a function of time, as well as the depth and time averaged concentration will be shown by Figure 4. 7b. If the time required for the grain to fall back to the bottom t^ is b greater than the wave period (where the dominant crest force is responsible for suspending the sediment) the concentra- tion will be shown by Figure ^-.Ic, with higher depth-time averaged concentrations than the previous case. It is therefore apparent that the depth time averaged concentra- tion should be a function of the parameter This ^s heuristic argument should be kept in mind as a more for- malized approach is now used in an attempt to arrive at an important parameter for use in prediction of suspended sand concentration in the surf zone. It is important at this point to look at the findings of some investigations of suspended sediment in the surf 140 Figure 4.7. Sediments Suspension Due to (b) Long Periods, High Fall Velocities, (c) Short Periods, Low Fall Velocities 141 zone. Watts (112) conducted field studies using an elabo- rate continuous suspended sediment sampler from a pier. His results showed that the amount of sand in suspension was related to the wave height, or energy, of the waves for a particular test. In these experiments, and some by Fukushima and Mizoguchi (113) using suspended samplers made of bamboo poles, the vertical distribution of suspended sediments was also measured. These data showed that the amount of sus- pended sediment can be fairly well described over the ver- tical by plotting concentration versus elevation on a semi- logrithmic plot for the case of spilling breakers. Fair- child (IIM-) using field equipment similar to that of Watts and measuring suspended sediment from piers found results similar to Watts. In most all cases where plunging breakers were not encountered, the concentration of suspended sedi- ment was found to be exponentially distributed over the vertical water column. Fairchild (114) also found this exponential type distribution in similar studies conducted in a wave tank. Hom-ma and Horikawa (115), Shinohara et al . (116), Hattori (117), and Kennedy and Locher (118) have all found this exponential type distribution under shoaling waves which is suggestive of a constant "sediment exchange coefficient" in the vertical. This appears to be particu- larly the case for spilling breaking waves (116). Reviewing the equilibrium exchange equation for sediment in the 142 vertical water., column- as given Ro.ose (64) where (using time averaged values as denoted by a single bar over the quantity, and primes as fluctuating quantities) w C(z) = -£ = V C (4.145) s s dz p where £ = vertical sediment exchange coefficient analgous s to the momentum exchange coefficient or eddy viscosity, and is the sediment particle random velocity. If the exchange coefficient £^ is independent of the z direction, then upon integration an equation can be found of the form C(z) J ^s^^-^a^ exp < - Ciz) a. £ S (4.146) where C (z ) is a concentration at the level z above a a the bottom. Since z is an arbitrary distance it. can be a assumed that z^ corresponds to the top of the bed load zone (Z = 0) and the concentration C (z) can be depth integrated from z = 0 to z = D giving _ -w D £ C = C (0) (1 - e — ~) —fr (4.147) £ w D s s A method for attempting to rationalize out a parameter of importance in suspending sediment ( "entrainment" ) to a level near the bed (in order to empirically obtain a value of ^(0) and hence C) is to consider the forces on a grain of sand in the near vicinity of the bed about to be lifted out into the flow (see Figure 4.8). 143 The lift force on a given grain "L" can be expressed as S L Z 4 and the gravity force weight G action on the particle is s expressed as 3 3 = (p^ -p) g = pCSg - 1) g (4.149) Now the ratio of lift to weight is 2 L 3C, u, s L hw G_ " 4CSg - Dgd (4.150) As u^^ is oscillatory, the time average value of the lift to gravity is L 3C, u, ^ G- - 8(Sg - Dgd C4.151) s In linear wave theory the maximum velocity (just outside the wave boundary layer) is u, (-D) = I ( ' I ,r. ^ (4.152) hwm T smh kD and the lift to gravity ratio can then be expressed as r 3c, s _ L H IT T sinh kD ^ =^ (4.153) G 8KSg - 1) gd o As grain size pertains to spherical particles in the above formulation a more general formulation applying to non- spherical particles can be made by expressing the grain size 144 Figure 4.8. Vertical Forces on a Grain Particle About to be Lifted from the Bed 145 in terms of its fall velocity w by a balance of drag force to submerged weight of the grain. This balance gives the following equation for a spherical particle falling at its terminal velocity w = (i - 1^ gd)l/2 "s ^3 ^ (4.154) where is a drag coefficient. Now equation 4.15 3 can be rewritten as 2C D H TT T sinh kD (4.155 ) w and thus the parameter of importance in predicting a concen- tration C(z ) in the vicinity of the bottom is the ratio of a ^ bottom horizontal water particle velocity to sediment par- ticle fall velocity IT C(z ) = function ■ a H T sinh kD w (4.156) In shallow water, sinh kD Z kD and the above expression can be reduced to C(z ) = function 3. w ^D s Note that in equation 4.15 6 the parameter H (4.157) heuristically arrived at previously, appears. 146 To obtain the relationship of e to wave parameters a s _^ look at the basic physical mixing phenomena is again neces- sary. The mean product V' C can be correlated and the ST degree of correlation expressed by a correlation coefficient 3^ which can be defined as 3 = 2 (4.158) / ,/ 9 7 C ' V P Then analogous to turbulent flow theory (54) the flue- ' tuatmg concentration y C ' can be related to the mean concen- tration gradient as C'^ = (4.159) where 5-^ = a mixing length analogous to the mixing length of Prandtl (64) for turbulent flow. Using the above two equa- tions a new equation for the turbulent fluctuation can be found as = - ISi! V'y^ • h§ (4.160) where the minus sign expresses the fact that the transport is in the direction of decreasing concentration. The product ^^\J ^ ^'^ is known as diffusion coefficient for sediment and is given the symbol e^. 147 where l-^ is in length units, is dimensionless , and ]J V^^ is the root mean square fluctuating velocity component. It will now be assumed that in spilling breakers the turbulence level is constant over depth and that the mixing velocity scale is proportional to the turbulence velocity which, in turn, is proportional to the average energy dissi- pation in the water column averaged over depth. I— (Ec cosa) - ^ ^ (4.162) Using shallow water assumptions, linear wave theory, and assuming as before that wave angles are small the following equation can be found Now, the mixing length l-^ will be assumed proportional to H which is the vertical distance a water particle on the surface travels through during the passage of a wave. H-^ " E (4.164) The sediment mixing parameter can then be found from equations 4.151, 4.16 3 and 4.164 to be (with = constant) £ = constant ^ ;g '^^^^ (4.165) 148 The data to be used in the assessment of the parameters C(0) and e come from field studies done by Fairchild (119) and Kana (120). In the study of Fairchild, suspended sedi- ment concentrations were taken throughout the surf zone from a pier at Nag's Head, North Carolina. The concentrations were measured using a tractor mounted pumping device attached to a pipe manifold extended over the pier into the surf ione. Time average concentratioixs were taken and the concen- tration was measured as the dry weight of suspended sediment filtered out per weight of fluid pumped. In the measurements of Kana, the concentrations were obtained at a location on the beach near Price Inlet, South Carolina by individuals stationed throughout the surf zone using water samplers activated by the individuals. The devices are described in (120). The concentration is then the weight of suspended sediment per weight of water sample. Data used consisted of depth-concentration profiles for spilling breaking waves in the surf zone. Wave parameters were measured by a wave gage at the end of the pier in the case of the Fairchild data, and by visual wave estimates (using wave poles and stopwatches) in the case of the Kana data. Unfortunately beach slope at the data collection sites was not taken. In most all cases the data gave reason- able results with the assumption that the mixing parameter is constant throughout depth, that is, the data plotted up 149 as straight lines on a semi-logrithmic plot of concentration versus elevation above the bottom. Data sources and data are given in Appendix B. The data used were for spilling break- ers although in a few cases it appeared that the concentra- tions were scattered with depth (not straight line) in some of the questionable type breaking wave data. In these cases it was assumed that the data scatter was due to plunging type breakers and the data were discarded . From the data plots and the sediment property analysis, values of were calculated and values of C(0) obtained from plots. These values were then plotted as functions of the relationships provided by equations 4.157 and 4.165. The results of the analysis are shown in Figures 4.9 and 4.10. Figure 4.9 shows that in fact the sediment mixing coefficient is well represented by a linear trend with the parameter - — g — which is a measure of the turbulent energy dissipated throughout the water column modified by a mixing length Cin this case wave height). Figure 4.10 provides a functional relationship between C(0) and a par- ticle lift to weight ratio parameter. At lower values of the lift to weight ratio parameter the concentration C(0) drops off sharply while at higher values there appears to be an approach to a linear trend. More data would be needed to confirm this trend outside the limits of the existing data. The bed load efficiency factor is found from argu- ments postulated by Bagnold (45) concerning a movable 150 151 + S a. Q i£2 X $ o c o •r- +-> fO S- s- -(-> 0) +J OJ O) u 5= c (13 o S- o ns a. -u o OJ ■r— -P •r- -o q: 00 <u 0) +j OJ o CQ Q-+-> •r— <+- J= _i O T3 •r— +J n3 ^ o ro ^ •1— Li- eu 152 boundary. Bagnold (US) pos-tiilates a continuous moving grain carpet relative to a stationary bed. Assuming a velocity of the moving carpet = the shear on the carpet (assuming 2 fully turbulent flow) is t ~ (V-V^) . The transporting work 2 done is then xV " V CV-V ) . This function has a maximum c c c value when = ^ hence the maximum transport efficiency is TV , c _ 1_ ®c " tV 3" Bagnold then postulates a similar slip of the actual grains within the moving grain carpet relative to the carpet which introduces an efficiency factor e = jj— such that g e e, = e • e = Again the work rate on the moving grains b c g 3 ^ n ' has a value V, (V -V, ) where n' varies between 1 and 2 b c b according to the local grain Reynolds number (V^-Vj^)d/v, being 1 in the Stokes law region and 2 for large grains . As before the work rate has a maximum when V, /V = b c l/(n'+l). The exponent n' for a given grain size d and slip velocity of grains V^-V^ can be obtained from the slope of the experimental log curve of the sphere drag coefficient versus the local grain Reynold's number. Values of e and e^ can then be found corresponding to a mean flow velocity V for a given grain size. As per Bagnold (45), the values of e^ and e^ are given in Figure M- . 11 over an extended range of values of V. 153 The dynamic Bedlload friction factor tan^) has been mentioned previously to be dependent on solid/fluid param- eter analogous to a fluids Reynolds number. Bagnold (121) found that tan<i> was dependent solely on a parameter d /v^ ~ where as before, X = ratio of the mean diameter ^ ^ ^ X of the solids to the mean free distance between the solids a-nd p - d-en-slty of -the solids. The parameter R^ is thus analogous to a "shear" Reynolds number of the form ^ _ d /t/p _ du^ shear ~ y / p " v In experiments using grains of equal density to the fluid being sheared, Bagnold (121) measured normal and tangential stresses due to the grain collisions in the annular space between two concentric cylinders as a function of the parameter R . The results of this series of tests s are given in a plot by Bagnold (121, 4-5) and reproduced here as Figure 4.12. The value assumed for X was equal to 14 which was the limiting concentration at which a sheared array of solid grains was found to cease to behave as a Newtonian fluid and to begin to behave as a rheological "paste" (121). 154 o d o d 155 CVJ T3 CM o 8 <M O O g o CO O d d a o UDj rO d 156 4.4.5. Model Comparison with Laboratory and Field Results In a comparison of the model with existing sets of data for sand transport, four data sets were chosen, two sets of laboratory data and two sets of field data. Although there are many existing data sets as discussed in Chapter II, other data sets failed to include vital wave parameters necessary to the model comparison; most typically, breaking wave, .ajigla was^nat given -but ojily the offshore wave angle measured at the wave generator (in the case of laboratory data) or in deep water (in the case of field data). It was felt that transforming wave parameters to breaking condi- tions involved inclusion of an additional uncertainty factor in the data and was therefore undesirable from the stand- point of assessing model validity. The four data sets chosen for model assessment include: the Shay and Johnson tests (laboratory); the Fairchild tests (laboratory); the Watts tests (field); and the Komar tests (field). Parameters were calculated in accordance with equations, graphs, etc., of this chapter and the calculated model parameters are listed in Appendix B. Wave heights are assumed as root mean square wave heights in the case of all data. In the Watts data, model parameter values were averaged on the basis of 0.5 to M- days of wave records, the time interval in which the sand transport values were given. As was shown earlier, the fact that the longshore current is not linear across the surf zone causes the value 157 of in equation 4.126 to be different than the postulated value. Also, due to the randomness of waves in nature, it might be expected that a considerable portion of the break- ing waves might be of a plunging type though the average wave conditions are of a spilling breaker type. Kana (12 0) has noted considerably higher concentrations of sediment in the plunging typ^ breaJcers which would predict much higher suspended load and, in turn, higher total sand transport. For these reasons, the sand transport model correlation was made plotting the immersed weight sand transport I,^ with the factor XP^- The plot of all four sets of data which cover-- four orders of magnitude of the 1,^^ range is given in Figure 4.13 by the best fit equation: = 2.82xPj^ . (4.166) A more detailed plot of the Fairchild data alone is shown in Figure 4.14 where the best fit equation to Fairchild data is given as : = S.OlxP^^ The best fit equation to the field data of Komar and Watts is given by the equation: I, = 4.24XP, and is shown in Figure 4.15. 158 Figure 4.13. Sand Transport Model Relationship 159 I^(#/sec) Figure 4.14. Best Fit Sand Transport Model for Fairchild Data 160 Figure 4.15. Best Fit Sand Transport Model for Field Data IGl As assumptions .liav.e ieen incorporated into the model which cannot be checked out by the present data (i.e., roughness, spilling breaker assumptions, monochromatic wave assumption) the model appears to provide a reasonable fit. Additional supporting evidence for the bed load component of the sand transport model is given in the work of Barcelo ' (82) who ran tests as noted in Chapter II in which he noted that no component o-f -suspended load was present below a critical wave energy level. For wave energy levels below that critical value (bed load), the sand transport was found to be proportional to the square of the sine of the breaking wave angle x 2 , that is : Q. ~ sin^(2a- ) Conclusive proof of the postulated model must await detailed laboratory testings in large wave basins with measurements made of all necessary wave, beach, and sediment parameters. CHAPTER V APPLICATION OF SAND TRANSPORT EQUATION TO CALCULATION OF LITTORAL DRIFT USING SHIP WAVE DATA 5 . 1 Application of Model In this chapter the equation derived for sand transport will be applied to compute littoral drift using a source of visual wave observations made from ships. Prior to application of the sand transport equation it is worthwhile to review some of the important assumptions inherent in the method to be used. CD Linear theory is valid for the wave transformation process and the wave energy present in the wave system; (2) Assumption in calculation of a "friction" coefficient are not violated Csee Reference 59); C3) Bottom topography is composed of straight and parallel bottom contours; CU) No drastic changes in the bottom profile are encountered in the shallow areas seaward of the breaker line up to the beach; (5) Adequate sources of sand are available for trans- port; Item (1) refers to the mathematical formulation of the problem and its relation to physical reality. As has been 162 163 previously noted, this assumption is reasonably good up to the region of breaking waves where it departs from the actual situation. Item (2) assumptions will be discussed later. Assumption (3) is necessary for the simple applica- tion of Snell's Law of Refraction used in this report and does not require a monotonic decrease in depth toward shore, but only the aforementioned relationship between bottom contours. Assumption (4) is necessary due to the use of offshore wave conditions for the computation of longshore energy rather than nearshore conditions. Thus, rock or coral reef might cause a large dissipation or reflection of energy before the wave reaches the computed breaker zone, which would be incorrectly included in the estimation of expended wave energy in the surf zone. An additional assump- tion inherent in the same transport model is Item (5), the availability of sand to be moved- This is dependent on the geologic processes acting in the area, and the natural or man-made conditions present. In some shoreline areas of Florida there is a lack of sand, predominantly in areas having extremely low wave energy. Rivers, inlets, jetties, groins, seawalls, prominent headlands, and submarine ridges and valleys can also cause a lack of sand in an area downdrift of an obstacle. A lack of sand supply causes erosion and in turn a depleted sand reservoir, with less sand available for the transport downdrift of a barrier. 1614 The littoral drift , can -be e:xpressed in terms of a volume transport rate rather than immersed weight transport as noted in Chapter II by the following conversion I, = (p^-p) g(l-p^) Q, (5.1) Due to the present more popular method of expressing the transport rate as a volumetric rate, the results of this chapter will provide values of (volumetric transport rate) in cubic yards per day. 5^ 1:1 r Pata "Source The wave data used in the computation of longshore energy flux and consequent littoral drift in this report can be found in the U.S. Naval Weather Service Command — Suimnary of Synoptic Meterological Observations , Volumes 4 and 5 (122), hereafter referred to as SSMO. These volumes are a compilation of meterological and sea state observations taken from ships travelling through "Data Squares" defined by their latitude and longitude boundaries. The percent frequency of wind direction versus sea heights can be found in SSMO Table 18 for different data squares on a monthly and annual basis. The percent frequency of wave height versus wave period for both sea and swell observations can be found in SSMO Table 19 for different squares on a monthly and annual basis. Computations of sand transport will use the data from both of these tables. Necessary assumptions made in the use of SSMO data are presented and discussed below. 165 In the use of Table. 18 the., assumptions have been made that (1) swell waves are in the same direction as the sea waves, which in turn correspond to the wind direction; and (2) waves are propagating in one direction only, the ob- served direction, in any specific time interval. In apply- int Table 19, the assumptions are made that (1) sea and swell waves of the same period and height can be treated alike, and will not lose- energy to the atmosphere between the point of observation and the portion of coastline con- sidered; (2) no other wave heights or periods af^e present during the observation of recorded wave with a given height and period; and (3) all observations were made in" "deep water" (h>_2.56T^ in ft.) for the wave periods recorded. Correlation between the ranges of wave heights, peri- ods, and directions given in the SSMO data volumes and the corresponding values used in the calculations of drift can be found in Appendix II. Due to the nature of human obser- vation of waves, the heights and periods found in the data tables should be considered as-significant heights and periods . 5.1.2 Analysis of Wave Data to Compute Sand Transport The immersed weight sand transport rate as given by Equation M-.12 7 can be modified to provide the volumetric sand transport by the equation 5.1 which can then be ex- pressed as 166 YCS -1)(1-p ) (5.2) where the terms x and depend on wave climate parameters H^, T, as well as sediment fall velocity. The above equation represents sand transport in terms of one wave height, period, and direction, in a deterministic sense. Considering a continuously changing state of offshore wave .conditions-, heights-, .periods-, ..and directions , the total littoral sand transport would consist of average sand trans- port weighted in accordance with values of for represen- tative wave heights, periods and directions. Thus for continuously changing wave conditions , the total longshore sand transport as averaged over a time interval t* would be ^ ^ Jt=o Jt=o ^ The value ^ can be thought of as the fraction of time over which a specific wave having a certain height, period, and direction is being generated during the period t". Express- ing these results in finite intervals : f(H^,T,9) = frequency = ^ (5.4) and t=t* Q = I Q/t).f(H^,T,e) (5.5) t=0 1B7 where F =00 O T = oo e = 2TT ^ I I I f(H^,T,0) = 1.00 C5.6) H =0 T=0 9=0 o with 6 equal to the azimuth of the direction from which the wave is propagating. It is related to by the equation: a = 0 - 9 where 9 is the azimuth of the perpendicular to on n the shoreline (see Figure 5.1). For waves reaching the coast, the summation would be as follows with 9 = 9^ - ct^ and ranging from -90° to +90°; H „ 9=9 +5 0 T=<=° n 2 ^ 1 I I ^ f(H T,9)<1.00 (5.7) H=0 T=0 9=9-4 o n 2 Note that in the above summation, when waves are being propa- gated away from the coast, that no wave energy will be avail- able for sand transport. Therefore the total sand transport becomes Qo = I I l"" ^ Q (H ,T,9) . f(H T,9) (5.8) ^ H =0 T=0 9=9 -1 ^ ° ° o n 2 The value of f(H^,T,9) can be computed by means of SSMO Tables 18 and 19. From Table 19 a value of f-^g(H^,T) is ob- tained such that H =" ^ig^H^'T) = 1.00 (5.9) H =0 T=0 o From Table 18 a value f-,o(H ,9) can be obtained corresponding X o o to a wave height range in Table 19 such that 158 Figure 5.1. Definition of Azimuth Angle Monnal to Shore e^, and Azimuth Angle of Wave Propagation e 169 f,o(H*,e) = 1.00 (5.10) 6=0 1^ ° where the * represents the correspondence of in Table 18 to the same range in Table 19 . Multiplying these two fac- tors together gives the desired frequency as a function of wave height, period, and direction. ?(H^,T,e) - ?^g.?^g (5.11) By the use of Equation 5.8, the longshore sand transport can be obtained in cubic yards per day, as averaged over any given period of wave observations. As mentioned previously, the representative values of H^jTjB for the ranges given in SSMO are discussed in Appendix C. The procedure for the calculation of longshore sand trans- port Q,(H ,T,a )«f(H ,T,9) is as follows: X. o o o (1) Compute the onshore directed component of energy flux ' o c cos a. from deep water conditions, that 8 go o is, the representative conditions for given wave height, period, and direction ranges (where E = — ^) ^o 8 ^• 2 (2) Compute the quantity , a bottom frxction energy dissipation coefficient discussed in Chapter II, to a shallow water depth, h , outside the zone of 170 breaking waves by numerical integration procedure of reference (59) (along the coast of Florida this depth was normally taken as 10 feet). (3) Calculate the breaking wave angle by an equation based on Snell's Law and linear wave theory as postulated by LeMehaute and Koh (12 3) a, = a • (0.25 + 5.5 E/h) (5.12) bo o o - my Calculate- the -co efficxHut "x and the longshore energy flux P^^ (as modified by friction dissipation (59) where 2 2 P = Kj:,(E, c ,cosa, )sina, = K^(E_c _cosa^)sina, Jifbgb b b fogo o b (5.13) (5) Find f^g and f^g values in SSMO Tables 18 and 19 as mentioned previously, and calculate f=f^g'f^g. Calculation of is then a simple summation process in which the data must be put through a "filter" to elimi- nate all differential bits of sand transport with azimuth directions 0 that are less than -9 0° or greater than +9 0° to the coastline azimuth 6^. When looking offshore a positive values of = Qi^^^ is recorded for waves propagating from the left side and causing longshore sand transport to the right; and likewise, a negative value of Q^^ = is re- corded for waves propagating from the right and causing longshore sand transport to the left (see Figure 5.2). Summing the positive, negative, and total values of long- shore sand transport gives the quantities Qjj^+j ^Jl-' "^ilnet * 171 Figure 5.2. Relationship Between Direction of Wave Propagation and Direction of Longshore Sand Transport 172 Additional assumptions used in the preceding method of calculation which were not previously discussed are: (1) There is minor loss of energy through bottom fric- tion between h^, the shallow water depth at which is calculated and the breaking depth. (2) Computation of K^, , and involve linear theory and inherent assumptions. (3) i_s ca.lculated -usi-Qg a bottom profile perpendicu- lar to the stretch of shoreline considered rather than the actual profile over which the waves travel. Inherent in this procedure is an addi- tional assumption that the wave climate used occurs at a point offshore perpendicular to the portion of coastline considered. (M-) Data weighting of wave heights, periods, and directions is accomplished linearly using the centroid of the SSMO data square as the source of the offshore wave data (see Figure 5.3 for location of SSMO data squares). (5) Assumed friction coefficient f = 0.02. w 5 . 2 Results of Sand Transport Computations Littoral drift "roses" with annually averaged values of littoral drift in cubic yards per day have been computed using the SSMO annual data summary tables along sections of Florida's sandy shores. These are presented in Figures CI 173 174 through C26. Because of the large number of these figures they are located with the Appendices Section of this thesis. A littoral drift rose diagram for each section of coast considered gives the annually averaged total positive and negative littoral drift. Positive values of littoral drift refer to drift moving toward the right when looking offshore, and conversely, negative values of drift are quantities of drift moving to the. J. eft a^, noted previously. On the East Coast of Florida a positive value of drift would thus repre- sent Southward drift, while on the Gulf Coast, the reverse would be true; that is, a negative value of drift would repre- sent Southward drift on the Gulf Coast. The net drift values represent the difference between the Southward and Northward total values of drift with the direction of the drift indi- cated by its sign as described above. Although the littoral drift has been computed for coastline orientations ranging over 360° of the compass, in actuality, the coastline orien- tations range at most over 18 0° for any given section and have been presented showing the maximum practical range plus or minus 20° for local anomalies. As mentioned previously, these values of littoral drift are for stretches of coast exposed to the ocean wave climate as represented by SSMO data. They are not valid for bays, lagoons, or estuaries, where the shoreline is not exposed to a wave climate represented by the SSMO data. Also, they are not valid where local anomalies exist in offshore bathymetry. 175 5.2.1 Use of a Littoral Drift Rose Use of a littoral drift rose is as follows: (1) Determine the orientation of coastline at which a drift is desired. (2) Using the azimuth of the seaward directed normal to the coastline at the location, find the values of total positive and total negative littoral , ..drift assoaia.ted with thi.^ azimuth angle on the proper drift rose corresponding to the desired location . (3) Find the value of net drift as the difference between the positive and negative drift values. If the net drift value is positive, the net drift will be to the right when looking offshore; if negative, the net drift will be to the left. To demonstrate the method, values of net drift at Ponte Vedra Beach, south of Jacksonville are found from Figures 5.4 and 5.5. The azimuth angle of the perpendicular to the shoreline is 76°-30' as shown in Figure 5.4. Thus, the total Southward drift is 1600 cubic yards per day, and the total Northward drift is 810 cubic yards per day from Figure 5.5. The net littoral drift is 790 cubic yards per day or 288,000 cubic yards per year to the South. Limitations in the simple procedure for calculating drift values in the above manner will be discussed, taking into account some of the data limitations and data bias. 176 Figure 5.4. Azimuth of Normal to Shoreline at Ponte Vedra Beach, Florida 177 Annudlly Av*roij9d Totol Littoral Onff St. John's Rivor to St. Augusiltie lnl«t N Figure 5.5. Determination of Total Positive and Total Negative Littoral Sand Transport 178 5.2*2 Possible Sources of -Data EvjpQr .Qr ^a s In the SSMO data, possible sources of error include: (1) Human error and bias in the observation and recording of the wave data. (2) Absence of extreme wave conditions due to routing of ships out of bad weather. (3) Inaccuracies introduced due to the lack of swell direction data, (4) Inadequate resolution of wave data direction wise. (5) Inaccurate wave height recording due to wave observation in a strong ocean current. Error sources (1) and (2) are self-explanatory. In regard to (1), it has been shown that a large bias is intro- duced in the directional data due to the observer tendency toward recording of wave directions along the four cardinal and four intercardinal .points of the compass. This effect is seen in the littoral drift roses. It is felt that the bias should not significantly affect the results presented here though, since wave directions used in the- computations were reduced to the eight points of the compass in the SSMO volumes. If it is assumed that the waves were recorded to the nearest point of the compass (on an eight point system) , the maximum error between a recorded wave direction and its true direction would be 22-1/2°. It is recommended that values of drift in a range of azimuth angles 11-1/4-° to 179 i:he actual coastlin-e aziinu-th- i>e ■Tso-nsadeT-ed as the range of possible drift values, thus covering a 22-1/2° range of possible wave directional error. The original method of reducing the data from 36 points of the compass to 8 points of the compass given in the SSMO volumes introduced a skew of the data by an angle of 10 degrees clockwise. This has been compensated for in the littoral drift roses and offshore wave climate roses by shifting rose azimuth angles 10 degrees counterclockwise. As mentioned earlier, the lack of swell direction data, and distinction between sea and swell, cause the assump- tion to be made that swell waves are being propagated in the same direction as the local wind waves (which is the recorded wind direction) . It is unlikely that swell is always in the same direction as the local seas and this could lead to considerable error in the computation of long- shore sand transport. In regard to Item ("4), since long- shore sand transport is dependent on wave direction due to refraction process, the method of computing sand trans- port by using only eight points of the compass poses a question as to the magnitude of error possible in the results. It can be shown that the maximum error introduced by this approach as compared to spreading the energy evenly over all directions within an octant is 10 percent. 180 In regard to Item .( 5) ,.-.wave- iaights are affected by strong currents and have the tendency to steepen when propagating against an opposing current and are reduced in height by a following current. This effect is noted on the Southeast coast of Florida where the Gulf Stream is very close to shore. Due to the fact that shipping lanes run through and along the Gulf Stream, it is felt that many of the ob&erved waves approaching shore have recorded wave heights higher or lower than would be experienced on the shoreward side of the Gulf Stream in comparitively still water. This effect would cause the computed South- ward drift values to be higher than the actual drift values and Northward drift values to be lower. 5.2.3 Other Possible Errors or Bias in Analysis A possible large source of error comes from the assumption mentioned earlier that waves are considered to be propagating in one direction at a time. That is, it is assumed that when waves are moving away from the coast, there are no waves reaching the coast, and thus there is no sand transport at the coast. This is a questionable assump- tion since waves are known to propagate in many directions at the same time but analysis of such error is impossible. Other possible sources of error involve the computation of the friction coefficient and the violation of Snell's Law with regard to the bottom contours . 181 .The modification of wave height due. to friction effects as the wave propagates across the continental shelf has factors which could contribute to inaccuracies as follows; (1) Friction coefficient (2) Method of taking profile for a coastline section (3) Neglecting friction beyond a certain depth, = h The friction coefficient used in this study was constant, ■equal to 0.02, but is known to be a function of bottom roughness (as noted in Chapter II), which in turn depends on wave height, and water depth. Thus, friction is not constant but varies with time. A sensitivity test was done using three friction factors: 0.01, 0.02, and 0.0 3 for the location of coastline which best represents an average profile from Fort Pierce Inlet to St. Lucie Inlet to compute values of drift. Assuming that the friction factor 0.02 is correct, a value of f=.03 gives drift values approximately 20% lower and a value of f=.01 gives drift values approximately 22% higher. The sensitivity would be much greater on a broader shelf width as in North Florida on the East Coast, and much smaller on a narrow shelf width as encountered in the south- ern limits of Florida on the East Coast. The method of taking a profile perpendicular to the stretch of shoreline considered leads to high values which would tend to overestimate the wave height and long- shore sand transport since refraction effectively causes 182 waves to travel over a longer profile than the one used. In view of the fact that locations of the individual wave data observations are unknown, the method of using a profile along the perpendicular to shore seems a reasonable approximation though. Assumption (3) which was mentioned earlier is made because the value of is based on linear wave theory, and,, near breaking conditions in shallow water, the wave form no longer corresponds to linear theory. In addition, the beach is in a dynamic state at shallow depths which would make assumptions regarding bottom profile in this region invalid during part of the year. 5.2.4- Other Potential Sources of Error Other factors which certainly have a bearing on littoral drift in an area but which were not accounted for in the present computations include : (1) Wind, tidal, and inlet refraction effects on littoral currents and corresponding drift. ■ (2) Sheltering effects of reefs, rock outcroppings , large submerged sand ridges, etc. (3) Interference in littoral regime due to jetties, inlets, rivers, sand sources, sand sinks, etc. Factor (1) has been found to be of major significance in some studies (12'4). 183 +J m Q o S CO oi bO C •H m t3 (U ■P 3 a e o o M (U dj C •H bO C w o CO a, u o u >, <U -M m e •H ■P CO W o •H O 0} u (U CO ra > 10 3 C c < u m o •H 3 U C x: x: 4= j::; x: .c .n xz - x: ■p-p-p+jp+j-p-p+j-p-p ooooooooooo CQCOCOWCOWWC/DCOUDCO CO o to ( — 1 C iH iH jH (J> CN H CO CN tO CN LO C30 H d" LO J" rH cn CO CO < — 1 CM rH r-\ CO CO CO CO +-' 1 t 1 1 1 1 1 1 1 1 1 4h H 0 f~ J" rH LO H CO CN rd A iH O cn CO CO CD CO CO \ CN iH CM CO H CM CN CO (—4 >H * — ' - -H U iH cu Q) w tiO rd *D C*^ CO o CJ CO CO r— 1 CN CO IT CO LO cn CO •H LO cn J" H LO CO LO CO CM CN CM CO CO CN CO O c <I ■H O ■H ^ x^ ^ sz ^ ^ -P -p -p ■p P +j +j 4-' ■P ■p O 3 3 3 3 3 3 3 3 3 3 3 O O o o O O O O O O o CO CO CO c/: 00 CO CO CO C/3 C/0 CO *H Q 0) CO 4J o n3 + -P u -H 0) Q O o o o o O o O O o OJ 0) o o o LO LO CO CO CM LO H is: a, LT) 1 Ln CO 1 1 CN 1 CM CM H o o o o CM +j -p 0! 0) H +J C C u 0) +J 0) M 1— 1 o P" (U P> P" > cu c H 0) c (a 0) c H 1— 1 C r-H o OS c o C C M G •H o •H M 1— 1 1— 1 P) o CO ■P o xi K, CO H C CU O O o c 3 0) rd OJ <u •H u U O •H !^ ■H •H O o O hJ ■P o 3 0) P> 3 ^ C3 < (U > M CO 03 o (T3 03 +-> (1) H ■H c G .a rH -P +j o 03 0) o P 03 •H < 00 CO &- O CO CO a: 0) o ill P" CO C <a w (0 ?H iH O u u 0) 03 > x w P" O CU e 03 J3 3 o CO 4= ^ P" P" 3 3 O O CO CO 00 CO CO LO CM LO CO CO o cn CO CO CD H cn 00 o cn LO o CO o cn H H I I I I I I I LO O r-l '"^ LO • r- rH . J- CO w . . CO CO LO z^ CO LO :i CO CN cn cn LO CO ^ :^ rH rH CD r~- CM CO P" P" 3 O O CO s PJ pl 3 3 O O CO CO O CO CO CO LO r- I I o CN O J3 ^ ^ ¥> V ^ fn 3 3 O O O S: CO CO CD CN O O CO CM J- O O CO rH pj G CO SZ (0 T3 CO 03 O rH C CO O CO 03 03 O CO Be et l-H Si CPl CO rH 03 1— 1 u rH 03 CO C ■H 0) 3 Ph U 1-H CO SU 0) P CD 0) cn 03 03 G >, 0) 03 S 3 :s U O s O P-. CO u 0) X) •H 03 03 m fn G C CP CD O O p C fn rH CD > 2: < &-1 184 +j +j +-> CO W M Q) OJ 0) 3: S 13 03 o CM O C-l CO CO I I I ^ :^ in CO CO C7> CN CM iH ^ CN I> CD LO 04 CN CM ■P +J -P CO CO w 0) (U (d S S W c o •H •p o OJ •H Q) 'X3 O o o <u CO c <u CO •p •H CO o &. a. o <u x: IT) ITS O ID CD ID H 0) V — -3 ■p CO < to CO o CO CO rd CO a. Ip 03 H cu rO CO a H to a o O [0 O Oi •H to 0} CO ■p C m 4) 0) to Oi w 4m •P to 0) +J 03 O •P T3 C CO CO ■P 0) 03 P4 185 The sheltering effect of reefs and rock outcroppings is certainly a factor affecting littoral drift along the south- east coast of Florida. Many rock outcroppings and reefs exist in the littoral regime and definitely influence drift values. In places such as Cape Kennedy where a large underwater sand ridge exists, the drift pattern is altered by the sheltering effect of the ridge which prevents some northeasterly wav^. from -reaching the .southern shore and some southeasterly waves from reaching the northern shore; thus, to an extent, the ridge tends to be a self-perpetuating littoral barrier. Jetties, inlets, rivers, submarine valley, etc. all influence the pattern of drift to alter it from the idealized model used to compute values of drift; these influences must be recognized when applying drift values derived by the approach presented. 5 . 3 Comparison of Calculated' Littoral' Drift Rates W3.th Previously Estimated Values Comparisons of the present study results with estimated values of net drift compiled by the U.S. Army Corps of Engineers are summarized in Table 5.1. The Corps of Engi- neers values were determined by various methods which include analysis of dredging records, volumetric surveys, and pumping records at existing by-pass plants. Computed values of drift by the present method provide both an "expected" value 186 of drift and, to illustrate the sensitivity of drift to coastline orientation, a range of drift values which encom- pass +11 1/4-° span of azimuths to the actual coastline azimuth, 8^, at a given location. Total positive and nega- tive drift rates with corresponding ranges of values are summarized in Table 5.2. Note that some of the comparisons presented in Table 5.1 and Table 5.2 may be misleading in that they are ex- tremely close to Corps estimated values when the assumptions involved in the program to compute drift are possibly vio- lated. One such location is Fort Myers Beach on the Gulf Coast. The computed value of net drift is 21,900 cubic yards per year in a Northerly direction which is extremely close to the Corps estimated value of 22,000 cubic yards per year. The assumption of parallel offshore contours is violated here though, and, refraction of waves from a North- easterly direction is undoubtedly much different from what the simplified analysis based on Snells ' Law would compute it to be. Refraction of N W Waves off Sanibel Island would tend to create a complex nearshore current situation with the probable direction of drift being North even if the wave climate and ideal bottom topography would normally tend to create a Southerly drift. It is suggested that use of the results presented in the littoral drift roses be carried out with a knowledge of 187 • cn CO • bo G ■H (U CO ■H >) 0) ■P 0) s ■P o nJ B •H +J ■ W H CO (0 CO (U +-> to ■P • ca U a +j <u CO c to -P •H fn C O d) •P CO w +J 0) •H U J &. o H 0) w to x: ■p +J ?^ o 0 o •H bO Id to 0) T3 OJ lO <U > rH ■P < 3 to o s H H •H to to ■P 3 a CO c w C tn < to O C to c O CQ •H 0) U 0) to c S M 0 c a w to (0 o o • O H H ,- — s a 0) c •H • Q) +J CD o CO +-> to to W Pi H to +J +-> o o •H CO El T3 O 0) H in <U ■P CO H 10 +J O O w +J 0) CO • o O C •H • 0) >, to 4h • •H CO CO • o a (7D ro w to ■p u o o tU >i ■p to u ■P Hh . •H CO O >- f4 >1 <D +-I O It u ■P o O w H CO • o CD c •H • CD >, ■P to U ■p •H CO U C o •H 4-> to o o 0) c to o J- 0) C to o J- o CM cn in H CN CD H cn r- 00 <y> CO CD CD (J) J- try ID' TO lO 1 in =i- 1 IT) 1 t>- 1 UD 1 CD 1 to 1 o 1 OD H 1 H CD 1 CO cn H J- CD (X) H CD CO J- m CN CM J- in CT) CO CD CO in r~ CD CD CO CO J- in CO J- CN r- cn CO CO cn cn cn CO CO in in in o o tn in rH J- CD O CO .H in o CD H cn ca o t> in CSI O CN CO CO CN 04 CN CO 1 CN CN ^ CM 1 CJl 1 H 1 1 CO 1 CN 1 in c-^ 1 H 1 o 1 CO 1 cn CD CN o CD i> =s- CO H C71 CO i> CN CO r-i CN CM CM CN CM o cn CN o CN CN CO u 0) > Pi o c o ■p -p CO CO tu c •H -P CO 3 0) bO 13 o c o p4 CN cn M CM o o o o o o CO CD CD o o o o o o H H H !h O -P ^ tu to c K M H G lO to !h -H (1) -P > CO rO to G ^ tO <U (l; CO •P 0) C <I) c O ^^ • O -P cn r-i c -p o lO CU o G to u -p CO G 0) w T3 to r-l O W) ^ ?^ u CD ro > K o • CO cn CO cn CN CO CN CD CO CD CO ■H to 1 rH-rH-H 1 1 1 H 1 H 1 1 CN CO r- cn CO in o CO o H CO 1 — i 1 CM J- CD CO H CO H in CO in rH iH in • - 1 1 in CO O o ^ CM in in OJ CO 00 H H rH in cn r- CN cn H r- o in rH CO CO CD rH H H c- 1 in 1 H 1 in 1 o 1 in i t> 1 rH CD CN CD CO in H CO CD in in (D m CO CO CD iH rH CO CO CO CD CO O CO CO o zt CO O rH co CO 00 CD cn H CN CM CO CN CN H rH rH I I CO CO p to O -H +-> CO to o o Mh rH c:3 0) o o to CD to pq CO n3 C-i ■P CD CO G U tH (U >, <D T3 G to rH CO 10 •H C o U • G O -P 0) C9 > CO to to S Pi a < T3 CO C CO to to H Ph m p to CQ U lO IB <U dJ rH H CJ 188 J- c- to CN CM CM o cn cN H O O iH H r-H cn m c~~ H CM o H H H H CD CD O) OJ o CO jj- I I I 03 iH H in ID CN CO CO CO rH CD H U3 CO ^ CO (0 +-> w < m M 0 M o n3 M M-i to CU rd to M o 0 O rO x) O p.. ■H 03 <U T3 Kl ■M C CO a, 0) OJ 03 3 a< W 189 the assumptions present in the'study such that one is not misled by the seemingly good comparisons as given above which may be fortuitous . Along the Atlantic Coast, the SSMO data confirm the Corps estimates of net Southerly movement of sand, and on most of the lower Gulf Coast the data confirm net sand movements in a Southerly direction. Along the Panhandle the net movement is in a Westerly direction which also agrees with other studies. Except for certain anom.alies in drift directions due to coastline orientation, the over- all trends confirm past observations with regard to direction, although magnitudes are different. Reason for the extremely high values of drift computed in Southeast Florida are not known at this time although the author speculates three possibilities : (1) The effect of the Gulf Stream current on wave height observations as mentioned earlier. (2) Effects of the Bahama Banks. (3) Resolution of wave data into large data squares rather than smaller squares where overall offshore conditions are the same. An interesting observation was made in this study with regard to null points in net drift. By viewing either the net drift diagrams or the total positive and total negative 190 drift diagrams, it can be seen that two types of null points exists in the drift regime. In Figure 5.6(a) a "Type 1" null point is shown for a portion of a typical total drift diagram. Assume first that an island exists such that its original orientation conforms to the null drift point (total positive drift = total negative drift), Figure 5.6(b). A perturbation in the system such as a storm, or the building of jetties ^at ends xsf.. the. .i^lan.d could cause the sand to be shifted to a position shown in Figure 5.6(c). In this case the net drift on the right side of the island would now be to the right while the net drift on the left side of the island would be to the left. Thus the overall effect of the perturbation would produce instability in the island, with the net result that the perturbation would increase and eventually the island would experience a breakthrough as shown in Figure 5.6. The orientation of the Gulf Coast shoreline in Lee County, Florida, is found to be approximately characterized by Type 1 null point. It is noted that this section of coastline contains numerous inlets and has a history of inlet breakthroughs. Another area where this type of null point is experienced is the Gulf Coast near St. Petersburg. Islands in this region tend to be extremely concave and would have probably broken through by now if not for the extensive groin fields hindering the transport of drift in 191 ji"^ (b) Ideal Island oriented 1o null point -zero net drift Perturbation in system causes orientation of island with associated drift pattern (d) Instability leads to eventual breaitthrough Figure 5.6. Ideal Case of an Unstable Null Point 192 the region. It should be noted that a perturbation in the convex sense would also be unstable and lead to an increasing convexity. No cases of this type system were noted on the open coast but numerous such features are seen in bays due to this type drift system. The second type of null point is shown in Figure 5.7(a). An ideal island when oriented to this type of null point has a tejid-ency to sta b-ilxze... atae.l.f once ^ perturbation in the system drives it from the ideal state. Figures 5.7(b), 5.7(c), and 5.7(d) show the series of events leading to stability. Part of the East Coast of Florida is near this type of null point where a predominant tendency for few inlets exists. Many of the inlets (such as Sebastian Inlet which occurs very near a "Type 2" null point) have had a record of numerous closures after being cut. Of course, many additional factors, influence stability and instability in true physical systems such as the amount of drift supplied to an area, and the ocean tidal ranges. These additional effects may overshadow those discussed here. It is hoped that in the future this theory can be explored further. Some further insight into the "null point" type can be given by considering two different cases of offshore wave climate, a unidirectional wave climate (all waves from one direction) and a bimodal wave climate (waves coming from 2 directions, 18 0 degrees separate). For simplification the 193 (a)- Type H null point 3n (b) Ideal Island oriented to null point — zero net drift (c5 Perturbation in system couses orianlolion of Island witli associoted drift pattern id) Self-stobiiizing drift pattern leads to original isiond conditions Figure 5.7.' Ideal Case of a Stable Null Point 194 sand transport wi 1 1 .all .be .considered to .be of suspended load and the sand transport equation to be of the simplified form = Ep sin2a^ (5.14) where - — * W( S -1) (1-p "7 constant (5.15) g ^o ■ Now when waves are coming from a predominant direction as shown in Figure 5.8, a littoral drift rose can be postulated as shown in Figure 5.8 where it has been assumed for simplic- ity that Ep=1.0. As the littoral drift rose drawn in Figure 5.8 is a "net" littoral drift rose it is apparent that this type of drift rose applies to the type II shoreline, a stable shoreline. When waves are coming from the directions as shown in Figure 5.9, and each directional climate contains the same amount of energy, a "littoral drift rose" can be postulated as shown in Figure 5.9. This drift rose can be seen to be of the unstable type of drift rose which leads to pertuba- tions in the shoreline. This type of wave climate and drift rose is believed to be the cause of the large pertubations (like small capes) in the shorelines of many elongated bays. An example of such a bay is Santa Rosa Sound in the Panhandle of Florida where a number of such anomalous shoreline "capes" exist (see Figure 5.10). The types of wave climate expected 195 WAVE CLIMATE CONSISTS OF ONE DOMINANT WAVE DIRECTION i Q^-^ ECg sin 20 ^ y> y y y y Figure 5.8. Stable Type Littoral Drift Rose Due to Unimodal Wave Climate 196 Q^/v Ecg sin 20 WAVE CUMATE Bl MODAL FftRALLEL TD SHORELINE FOR AZMUTH SHOWN Figure 5.9. Unstable Type Littoral Drift Rose Due to Bimodal Type Wave Climate 197 198 in these elongated bodies of water are conducive to "littoral drift roses" of the unstable type where a long fetch along the axis of the bay or sound allows large wave energy compo- nents coming from large angles while little wave energy comes from directly across the sound due to a restricted fetch. This is the type of wave climate conceptualized in Figure 5.9. Previously no explanations for these shoreline ..shapes existed. 5 . 4 Comparison of Estimated and Observed Wave Climates To determine the reliability of the SSMO data and computed shoaling, refraction, etc. effects in the present study, a comparison was made using wave records obtained from shore-based gages. Data from step resistance wave gages operated by the Coastal Engineering Research Center were made available for three wave gage stations : Daytona Beach (East Coast), Lake Worth- Palm Beach (East Coast), and Naples (Gulf Coast). Wave data were obtained intermittently during the years of operation of these stations due to various storms damaging equipment or structures on which the gages were mounted. To avoid a seasonal bias in the shore- based recordings, a sample of data best representing the average annual conditions was used in each comparison. Table 5.3 shows the observation period used, the total number of observations, and the depths of these stations. 199 In regard to the SSMO data, certain assumptions had to be made with respect to the frequency of occurrence for wave heights and periods. Only the onshore directed waves were used for obvious reasons, which gave an extremely high frequency of "calm" conditions (H=0) at shore. It was assumed for the plotting of cumulative height curves that the sea state at shore is best represented by wave heights of JLess, than one -£oot wh.en of f ■shore directed waves were being recorded. Most likely, many waves greater than one foot would be recorded at shore during this time. This assumption gives a poor basis of comparison for recorded and observed low wave heights in which the majority of waves fall. In the cumulative distribution curves for wave period, the assumed frequency of occurrence of a specific wave period was assumed equal to the frequency of the on-shore directed wave (of a specific period) times one (1.0), divided by the total fraction of onshore directed waves . Cumulative curves of the plotted wave height and period distributions at these three stations are shown in Figures 5.11 through 5.16. The wave height cumulative curves show three sets of points with corresponding smooth curves drawn through them, one curve for the CERC gages, one curve for the deep water onshore wave climate as recorded by SSMO, and one curve for the SSMO wave climate as modified by the present study to the depth of the recording wave gage. 200 Table 5.3. Recording Periods of Shore Based CERC Wave Gages Used in Comparison of Actual to Predicted Shore Wave Climate. Daytona Beach, Florida (Depth of Wave Gage = 15 ft. MWL) February-December 1954 February-November 1955 February-March 1956 January -April 1957 November-December 1954 Lake Worth-Palm Beach, Florida (Depth of Wave Gage = 18.2 ft. MWL at Lake Worth and = 15.7 ft. MWL at Palm Beach) January -April and June -December 1958 Palm Beach 1161 observations January -December 1960 Palm Beach 2020 observations January -December 1960 Palm Beach 1687 observations January -April and June -December 1961 Palm Beach 1301 observations January -December 1966 Lake Worth 1751 observations Naples, Florida (Depth of Wave Gage = 16.5 ft. MWL) January -December 1958 1454 observations 1570 observations^* 1151 observations 234 observations 321 observations 304 observations "'Each observation is the significant wave height and period as deter- mined from a 7-minute recording of sea surface elevation measured using a step resistance type wave gage. 201 202 -a o i 203 as •a •r- o CP) 204 IT3 -a s- o 0) i- • 205 206 207 Since it is assumed that periods are not modified by offshore topography, two curves are shown on the wave period cumula- tive distribution curves, one for CERC recorded wave climate, and one for SSMO recorded wave climate with inherent assumptions . The curves show that wave heights of the higher energy waves are represented well by the modified SSMO data. Unfortunately , thojjgh, .p^iods are .poorly defined by the data source. Due to the large dependence of wave modification on wave periods, it is felt that an even closer correspondence to offshore observations might be obtained with improved period observations. CHAPTER VI ANALYTICAL MODELS FOR SHORELINE CHANGE 6 . 1 Introductory Remarks As important as the ability to predict sand transport is, the problem of being able to predict shoreline change is equal 1 y i mportant when evaJ.iL3,±ixLg .optimal location of build- ing in the coastal-beach strip. The effects caused by reefs, rock outcroppings along the beach and coastal struc- tures (revetment, groins, jetties) on the shoreline are important to coastal engineers . A predictive capability in determining shoreline changes due to longshore sand trans- port changes becomes of primary importance in the design of coastal structures also. If accurate predictions cannot be made for determining the changes due to coastal structures, then such projects may be doomed to failure. This chapter first discusses a heuristic model for determining an "equilibrium" shoreline (a shoreline along which the longshore sand transport is zero) under given average wave conditions and is applied to specific areas of Florida's coast where nearshore reefs play an important role in determinirtg shoreline shape. The model is then given a more sound theoretical basis using the longshore sand trans- port equation. 208 209 A second model (dynamic and analytical) is then postu- lated using the sand transport equation and the equation of continuity via linearizing the sand transport equation similarly to the work of Pelnard-Considere (125) only with different parameters. Analytical solutions to the partial differential equation can then be found and solutions for various coastal structures (e.g. jetties, beach nourish- ments) can.be found and noji-d i Tn e n .siona3J.zed solution graphs given. 6 . 2 Heuristic "Equilibrium" Shoreline Model Development It has long been recognized (126, 127) that the shore- lines of sandy beach areas protected by headlands follow the shape of a logarithmic spiral curve. The most prominent logarithmic shoreline features noted in the literature are associated with natural rock headlands on what Johnson (128) termed as "shorelines of submergence." Shorelines sheltered by natural reefs, shoals, and coastal structures can also exhibit this logarithmic spiral shape though as shown in Figures 6.1, 6.2, and 6.3, exemplifying that such natural shapes are not just commonplace on naturally rocky coasts but exist on most all shorelines with sedimentary beaches. Figure 6.1 represents a large shoreline indentation just south of a natural offshore "worm" rock reef on the east coast of Florida near Vero Beach where the predominant direction of sand transport is southward as shown in Chapter Figure 6.1. Longspiral Curve Fit to Shoreline at Riomar, Florida 211 Scale — KilomeJera 212 GULF OF MEXICO Depth Contours in fest. Shoreline and Offshore Depth Contours from Nautical Chart N.O.S. 856 S.C. I I L 0 0.1 Q2 Q3 Q4 05 Scale— Nautical Miles -1 — I I I 0 .1 2 3 4 .5 Scale - Kilometers Figure 6.2. Logspiral Curve Fit to Shoreline at Lacosta Island 213 o o CD ■o fSS Q. o > o s_ •1— CL. to o vo 0) S- 2m V. Figure 6.2 represents a large shoreline indentation on the lower Gulf Coast of Florida on La Costa Island. This section of coast is also believed to have a net southward sand transport as per Chapter V. A nearshore shoal appears to be the cause of the indentation to the south. Figure 6.3 shows that this phenomena is not limited to only small sections of shoreline but can be somewhat representative of a large section of coa^t -such as the Panhandle section of Florida west of Cape San Bias where the predominant direc- tion of sand transport on the beaches is westward (to the west of Panama City). As the logarithmic spiral is found to describe smooth sections of coast so well it might be reasonable to ask if such a fit is just a quirk of nature or whether perhaps there is a more justified reason behind the phenomena? Some insight can be gained by looking at the equation of the logarithmic spiral which is of the form R = e^^ (6.1) The above equation can be reformulated into an equation of the form: lnR2 - InR^ = k(e2 - 8^) = kAS (6.2) or R^/R^ = e^^®. (6.3) When A9 is a constant arc size the parameter R2/R-j^ is also constant for a given value of "k." By a judicious choice of 215 the value "k" and of the use of a given section of the log spiral curve, an unlimited number of "smooth" curves can be drawn and fit to curved shoreline features. Figures 6.1 through 5.3 have been plotted using different values of the parameter R2 / R-j^ . Sea shells also have naturally occurring smooth curves and can be found to fit a logarithmic spiral curve quite closely also (129).,. There is no physical justification for this shape though, and indeed it is fortuitous, as shell shape can be found to closely fit more physically reasonable shapes such as ideal airfoil curves (129). Justification for the log spiral shoreline shape as well., appears to be lacking from a physical point of view and hence a more physically descriptive system for the equilibrium shoreline shape needs to be considered as in the following heuristic model. The present heuristic model describes the shoreline sheltered by a headland (or on a smaller scale, a groin or offshore breakwater) using a wave energy diagram consisting of representative offshore ship wave height and direction observations. Offshore ship data ane summarized by the U.S. Naval Weather Service Command in a publication entitled "Summary of Synoptic Meteorological Observations (SSMO)" (122), as discussed previously and used for computing sand transport along the coast. 216 To describe the simplest case of the present heuristic model. Figure 6.4- is presented with an x-y coordinate system parallel and perpendicular to the existing overall trend of a large coastline segment. Note that stavting in this ahapterj the x-y coovdinate axis convention has been changed to aovres-pond to simitar notational convention used by othev authors. The y direction now refers to an axis yerpendi- oular to the. shorelin& .j^hil& the x axis now refers to the axis parallel to the shoreline . In the simplest case of equal wave energy flux approaching the coast from all direc- tions the wave energy directional distribution is described by a semi-circle as in the top portion of Figure 6.4. At a point "P" described by coordinates (x, y) in a sheltered section of coast, by geometric considerations shown in Figure 6.4 it can be reasoned out that the slope of the shoreline in plan view, must be described by the equation: dy , , $ V = tan (2") (5.4) if there is to be no longshore component of wave energy flux (which has been shown earlier to be proportional to H^'^^) striking the shoreline in its final equilibrium shape. It should be noted here that the condition of no longshore sand transport along the coast is the same condition as that generally assumed to give the logarithmic spiral shape shoreline by earlier investigators (126, 127). This is a static condition since no dynamics of sand transport are 217 Figure 5.4. Orientation of a Shoreline Segment Having Equal Offshore Wave Energy 218 involved in the shoreline shape. Thus only a final shore- line shape can be arrived at and not intermediate time steps. Upon expanding the equation in terms of $ it can be shown that : g . tan ( tan-^ y/x ^ ^5^5) which can be solved for y in the form y^ = Constant • xCl + ^1 + (y/x)^) • (6.6) This equation can be solved by iteration for a shoreline shape or solved graphically from the original differential equation as expressed in Equation 6.4-. The more complex case of this simple static equilibrium model occurs for the typical case of an unequal distribution of wave energy flux impinging on the coast. For example, off the east coast of Florida the average wave energy flux distribution from, the SSMO data is summarized in Figure 6.5. 5/2 This distribution of H is formulated from, tabulated visual estimates of ship wave observations taken by commer- cial vessels and the military vessels within specified latitude longitude grid sections in the previously mentioned publication "SSMO." By summarizing the wave energy from 5/2 given directions using representative wave energy value H multiplied by the frequency of occurrence of that energy level, the continuous energy distribution shown can be found as a function of a direction angle, "Q." 219 Figure 5.5. Wave Energy Rose for Ocean Data Square #12 Off East Coast of Florida 220 The magnitude of this energy flux level is given by the value "E*." As the specific units of measurement for this 5/2 wave energy are in (feet) , the given level of energy at a specific direction angle can be thought of as an (average 5/2 wave height) impinging on the coast from the direction 6 at any time. This is obviously a simplified assumption to the true situation where wave energy is from different direritlDxis .at different times o.f the year-^and at different energy levels, but this simple representation will suffice for the conditions of this model. Using the energy distribution so described, the total energy striking the coastline in both parallel and perpen- dicular directions to the x axis in the sheltered area of the coast can be described by the quantities P" and N"^ respectively where: •180 v-k'^ r ay -cosae - J E*'2(e) (6.7) 180 In the sheltered area of the coast the shoreline will take the shape to minimize the longshore energy component of the incoming waves, i.e., energy component longshore = 0. This condition is shown in Figure 6 . 6 and is represented by the equation 221 Ei_ = Energy Component Longshore =0 =-N sinIZ + PcosX2 (in x direction) Figure 6.6. Energy Diagram of Coastline with Equilibrium Coast Shape 22 2 longshore en-ergy compoxLent - 0 - -N"sinfi + P^cosfi (6.8) or w = = ai ^6.9) where the quantity P-VN'^ = f(0). Thus the condition of the shoreline slope at a point in the sheltered area can be ascertained for the given angle by two integrations of the offshore wave energy diagram described by equation 6.7. It should be noted that this model does not take into consid- eration any effects of diffraction and therefore cannot model the effects of wave action cutting behind a headland, but where a solid structure such as a seawall, jetty, or groin is the limiting condition, the model might be expected to be very realistic. A graphical solution diagram for the east coast of Florida where the general shoreline trend is considered to run directly north and south is presented in Figure 6.7. At vari-ous values of angle $ the slopes have been plotted and curves fit to the slope values. The curves so drawn re- semble logarithmic spirals quite closely. Looking again at the same east coast shoreline area which was shown earlier, in Figure 6.1, a curve from Figure 6.7 has been fit to the shoreline at Riomar, Florida, in Figure 6.8. As Figure 6.8 shows, the equilibrium shoreline curve fits quite well with proper adjustment of the x-y axis origin. 223 Figure 5.7. Solution Curves to Equilibrium Shoreline Equation for the East Coast of Florida 224 Figure 6.8. Equilibrium Shoreline Shape for Riomar, Florida 225 Equilibrium coastline curves for a general shoreline tending north-south on the Gulf Coast of Florida using a similar Gulf Coast wave energy distribution is shown in Figure 6.9. These equilibrium curves also can be approxi- mated closely by log spiral curves. Figures 5.10 represents the earlier case shown for Lacosta Island, now with the equilibrium shoreline orientation given. Here it was found jiecessary to shift the .a,:xj s o-f the equilibrium diagrams 4 3- 48° to get a good shoreline fitting. The necessity of this shift to provide a good fitting indicates that a shift in the energy distribution diagram is necessary for the "true" wave climate, i.e., the predominant wave energy comes more from the west than from the northwest of the wave energy distribution diagram. Figure 6.11 presents an equilibrium shoreline shape for the Panhandle Gulf coast of Florida using the wave height distribution from data square number 16 of "SSMO." The equilibrium shoreline gives an improved fit to the shoreline shape in this area over the log spiral curve fit previously presented in Figure 6 . 3 provided a shift in the energy distribution diagram of 21° is made. This suggests a "true" wave energy distribution shifted more toward the west than the distribution obtained from the ship wave observations. As a final use for this heuristic method of equilibrium shoreline shape, coastal structures such as offshore breakwaters 226 Figure 6.9. Solution Curves to Equilibrium Shoreline for Gulf Coast of Florida 227 GULF OF MEXICO I L 0 Ql_ 02 Q3 0.4 05 Scnle— Nautical Miies I — I — I — I— ; I I I I 0 .1 2 .3 4 .5 Scale — Kilometers Shoreline and Offshore Depth Contours from Nautical Chart N.QS. 845 S.C. Figure 6.10. Equilibrium Shoreline Shape for Lacosta Island, Florida 228 229 can. be considered to limit the a.va 1 1 able wave energy strik- ing a shoreline. The sheltering effect of the breakwaters at a point P(x,y) in the lee of the breakwaters is shown in Figure 6.12 where the wave energy reaching point P is limited between directions <l>^ and Again the equilibrium shoreline slopes are described by the equation: (6.9) where P* and N* are again calculated by integration of the wave energy diagram times the cosine and sine of the direc- tion angle 8 respectively. Once the slopes have been cal- culated, smooth curves through the slopes can provide a set of potential shoreline shapes. In the case that the shore- line has a long stretch of protected coast both updrift and downdrift (i.e., a large number of offshore breakwaters or groins are in use) and the potential sand transport is zero along the coast, the shoreline shape must reach an equilib- rium such that the net accretion seaward of the original shoreline is equal to the net erosion shoreward of the original shoreline as in Figure 6.12. From the given shape postulated, the indentation of the shoreline can be cal- culated and thus the effects of building too close to the shoreline provided for. 6 . 3 Analytical Treatment of Shoreline Change Model One can also look at the sand transport equation as providing a mechanism for determining the "equilibrium 230 c o c <D o w u> « o o Ik. a 49 "o *S a a w a a II II a <s <u +-> O ra +J -r- o s- • i~ o D. I— HI o on s- o +-I M- O +J </) <T3 O +J to LJ <D 4J 03 o CO s~ CD -P (C 3 •I- ^ d) <u S- S- O CQ x: ui (U S- S O •r- to S- XI M- •r- O ^ >1 cr UJ s- 231 shoreline shape." Once again the sand transport equation is of the form e, sin2a, e C, cosa, D D ^ s D D tantj) 2 w s (6.10) -r _ /V- <(l-K)tan3N - ^\ f ^ w with „ P. = — H c, sina, cosa, as before which is equivalent it 8 b b b to the following expression assuming is small and cosa^=1.0 I, = b!' H, ^''^ sin^a, + s! H, ^^^sina, (6.11) 5- 1 b bib b where - (Y ^(l-K)tang s % ,Y jg. w ^ (5.12) e c, s 1 w w s Now to proceed, it is necessary to linearize the sand transport equation in the term sina^ by incorporating some average value of sina-j,^ into the above equation, i.e., i=M (sina, ) = y f. sina, . (6.13) b 1 bi 1 = 1 where a multidirectional wave climate is now assumed and f^ = frequency of occurrence of breaking wave with wave direc- tion a, . . Then b" = B"(sina, )" and the sand transport bi lib equation for a multidirectional wave climate (time averaged) is M I„ = (b; + S*) I f.H.^^^ sin(9 - 6, . ) iZ.l^) % 1 1.^^11 s bi 1 = 1 - (Y i<(l-K)tane , ^s'^b ,y Jg. ^1 - ^^w f ^ ^8 232 where a, . = 9 - 8, . = breaking wave angle with shoreline and 9^^ = defined angle of wave crest at breaking 6 = defined angle of shoreline s ° This convention is shown in Figure 6.13. Now for the "equilibrium shoreline" shape which has been defined as the shape the planform of the shoreline will take after sand has shifted sufficiently to provide a condi- tion of 1^ = 0 along- all- parls ryf trhe shoreline, the above equation ' becomes for a point "P" I, = 0 = (B* + S*) y f.H.^''^ since - 9, .) (5.15) 1 1 1 . 1 a. s bi 2.-1 which can be expanded using trigonometric identities and solved for the angle .8 to equal M f.ri. smS, . tane = ^ = ^ ^ ^ ^ (6.16) s M y f .H. ^ cose, . . ^, i X bi 1 = 1 In the sheltered region of coast the above equation can be applied at the boundary with a source of wave data and integrated in a step wise fashion with tan 9 = ^ until the ° ^ s dx entire planform equilibrium shoreline shape is described. As the first method is more intuitatively clear (therefore easier to spot mistakes) the second approach was not needed but was described for more theoretical insight into the method only. 233 y ELEVATION VIEW Figure 6.13. Definition of Sign Convention for Shoreline Modeling 234 Again it is important to note that the method does not take into consideration diffraction effects although with a very detailed analysis diffraction effects could be con- sidered in equation 6.16 by incorporating a diffraction coefficient K^^ along with the wave height (e.g., H^K^ rather than H^). The diffraction process is discussed in Reference (130). HJae previx)us " eqiidJLibrduin shoreline" model took into account only the static shoreline concept (i.e., where boundaries must eventually comply with the condition that there is no sand flux across them) . It is apparent that to describe the dynamic process of sand transport a time as well as spacially dependent model must be derived. Using the total sand transport equation as discussed previously, one can find the sand transport in terms of volumetric rate as before K <(1-K) tanB r sin2a, e c, cosa, ^ n - c w , J b b s b b| ^l' WcS^-1) (l-p^)f ^ * \tan4) 2 ^ / § o w s (6 .17) which by making the small breaking wave angle approximation 2 (cosa^ = 1.0) and linearizing the sin term in the bedload M portion of the equation as before with f^sina^^ = (sina^)" the equation becomes Q. = F, (B,S ,p ,f ) (— ^ (sina,)'-- + -2-5-1 P. (6.18) 1 ' g^o' w [tantp b w J £ 235 The above sand transport equation can be coupled with the longshore sand transport continuity equation |X=0 (6.19) which states that the difference in sand flux into a control volume is balanced by the storage (accretion or erosion) of sand in the control volume. The coordinate axis chosen in the present situation is shown in Figure 6.13, to correspond to other type approaches of a similar nature (125) as men- tioned previously (again note the axis change from that of previous chapters). The value h^ represents the entire depth over which the profile changes and consists of the berm height above water level "hj^^" plus the depth below water to the limits of significant longshore sand movement "D^." An implicit assumption in the above equation of continuity is that the bottom profile is constant in form and moves continuously seaward or shoreward due to the volumetric change in sand transport caused by a differential in longshore sand trans- port. This effectively says that there is no loss of sand to the control volume due to onshore-offshore sand movements but that such fluctuations in beach profiles average them- selves out to provide a net zero profile change during the period of concern. Also any sand flux transport through the boundaries parallel to the shore when averaged over the 236 period of concern has a net zero value. Fluxes of sand in the onshore-offshore direction could be taken into account in the present model by an additional sink-source term for gain-loss to the beach profile but as no data exist;; for use in such a formulation it will not be included. The problem of onshore-offshore sand transport is in need of consider- able research which takes into account both theoretical aspects of the problejn.as weJ.1 as practical data collection but will not be pursued further here. Now the sand transport equation can be differentiated with the assumption of small breaking angles as before sina, ~ a, = 0 - 0, . , and an additional assumption that b b s bi' ^ other quantities such as beach slope and wave height do not change in the longshore direction. Therefore the continuity equation becomes t 8t 1 e, (sina, )" e c, b ■ b s b tan(j) w 1 where Now with = arctan 3y 3x e, (sina, )* e^c, b b _^ s b tH. and ^b^b tancj) 2 b . w 3 y 3x (6.20) (6 .21) (6.22) along with the assumption that (■^)^ ^'^l we find at ^h^ E <! ) ^ Vh^ . 2 3x (6.23) 237 setting (6.24) which is the assumed constant for a given set of wave and sediment parameters . The equation can then be reduced to the form of the one dimensional heat flux equation for which various solutions exist (131 ) . The value of K* can be referred to as a "shoreline diffusiv- ity" constant. • The first solution discussed will be that which was solved first by Pelnard-Considere (125) under a differently postulated sand transport model. This situation is that of a coastal structure across the surf zone (i.e., groin or jetty) which stops all sand transport. For the condition of no sand transport at the location x = 0 (the structure location) , the boundary condi- tion to be fulfilled is that = 0 , or in other words dv ... tanG^ = A second boundary condition xs round at x = " where y = 0, assuming the initial condition is that y = 0 for time t <_ 0 . The solution for this case (although with a different value of K" ) was found by Pelnard-Considere to be (6.25) 238 y = tanG, j— exp - ( — - — ) — erfc(— — ) 1(6. 26) ^ ^/^ 2/K^ 2/K^ 2/K^ where erfcC ) is the complementar'y error function and is tabulated in various mathmatical handbooks (132). The solution can be made dimensionless by dividing by the length = 2/K*t tan 6^. A solution graph for this spe- cific application is presented in Figure 6.14. The value of K* can be found • f ro m - equa-tion 6.-2*4 and the -resulting graph used to plot solutions of the shoreline at various time increments . Pelnard-Considere (125) also considered the case in which the structure had filled to capacity and bypassing began to take place. The time required for the structure to fill "t=t^ can be found from the previous solution for the ordinate x = 0 where Y = length of structure = K^t, 1/2 2 (— ^) tane, . IT D Pelnard-Considere (125) provides a second solution for times after the structure has filled to capacity. The boundary conditions for his second solution are that y = Y at X = 0, and y = 0 at x = =° for all times. The initial condi- tions are as in the previous solution y = 0 at t = 0 for X > 0. The solution to these specific boundary conditions are as follows: y = Y erfc( — ) (6.27) 2/K^^ 239 240 which can be made dimensionless by dividing the above equa- tion by the length of structure Y. The dimensionless solu- tion is presented graphically in Figure 6.15. Pelnard- Considere used a time = t in equation 6.2 7 such that areas of shoreline above the x axis would be equal at the time when the structure is just filled to capacity in equation 6.25, i.e., matched solution plan areas. In this manner -t^ was found to be t2 = t - G.38t^. Although the matched solution of Pelnard-Considere was not exact it appears very useful for practical application. Other applications of the same linearized partial differential equation 6.2 5 can be used in prediction of beach nourishment fill changes upon suitable evaluation of the "shoreline diffusivity" coefficient k" . Some applications are as follows: — Beach nourishment fill on a straight reach of beach. Fill exists from -x <x<+x and extends Y a a distance from the original beach. The solution for this specific case is as follows: y = \ {erf ( )(1-^) + erf( )(1 + (6.28) I 2yWT a 2/K^ "^a ^ The nondimensionalized solution graph for various fill times is shown in Figure 6.16. — Point source type of beach nourishment such as might exist at a location where a truck dumping of 241 ■■ - 1 1 ■ ■ 1 1 _l 1 \ i 1 ■ 1 1 1 ■ 1 - - 1 .... .:. 1 . ■ ' 1 1 1 - 1 __ : 1 ■ — — _ , 1 - [ - - _ , — — 1 ... 1 ■ 11 I 1 L . — . i—. ^.r — ' -z^ — "to — -to — ' — £=i — ^ — — . . 1 1 - — ' i — o ■ .; I.I — .. - - ■ ■■ — '1 1 — — — — , -n— 1 I — ^_ — 1 . ! 1 , _ — 1 -es— 1 U- — r-<r— : — ~ — _j : -tdr— ■ —J n : ■ ■ ■ 1 -n — r\ — O I , — . — _ — H- , — : _ ...ijj., —J — < , CO - — — '■<_. : h— 1 7 / 1 / t : — r- — — - — . 1 ' 1 A 1 ■ - ■ - 1 — ' .... -1 -! — ■ • ! ■ / 1 1 1 ~ t r — ■ — — ) ■ 1 1 . ■ 1 h— 1 1 ; -- ! - -\ - \ 1 ■ ■ ! 1 -1 — _ — — — 1 — i— 1 . CM ■Q -OJ -d d o i. O) +J M- < OJ s_ s_ =: +-> 4-J u u 33 r3 i. S- +-> +-> 00 +-> H- fO o "O 0) •t— TD OJ cn -C o re OJ a: 00 s_ o ■a > s- OO o 0) •r- o ^- •1 — -l-J o 00 o GO d 03 d d d 242 243 sand occurs or jdrag scraping of sand from the offshore occurs. The solution for this specific case (where the original fill has been assumed triangular in shape) is: Y r y = _£ j (l-X)erf (U(l-X))+Cl+X)erf(U(l+X))-2Xerf CUX)+ 1 , -U^d+X)^^ -U^(l-X)^ o.-(UX) (e +e -2e (6.29) where X and U (6.30) 2/Wt The nondimensionalized solution graph for various fill times is shown in Figure 6.17. For the case in which a beach nourishment has been placed on an existing beach but a gap has been left in the beach nourishment project (such as occurred in a beach nourishment project on Jupiter Island, Florida in 19 74), the following solution would apply which is similar to eq^uation 6. 28 . y = J \evfc — X ( (1-— ) ^a + erf c ( )(!+!-) a (6 . 31) The nondimensional solution graph for this specific case is given in Figure 6.18. — The case of a semi-infinite beach fill gives the shape of the beach fill end and its extent of 244 245 246 247 pjTogression down tJne coa^t with time. The solu- tion for this specific case is y = yjl + -erf i— — )| (6.32) The nondimensional solution graph is shown in Figure 6.19. CHAPTER VII SAND TRANSPORT AiND STORAGE AT INLETS As the present sand transport model developed is for ideal beaches where gradients in the longshore direction are zero, the model would not be expected to work at inlets, where in addition to complex bottom topography, the often large tidal currents play an important factor in the trans- port processes. Little is known about the specifics of sand transport at inlets (133) although generalities can be dealt with. It is known that many inlets act as sand traps by storing sand in their outer shoals (133). They become self- perpetuating features by their ability to jet sand (which comes to the inlets by ' littoral drift) to the outer shoals where the sand is either cycled back to the shoreline during heavy wave activity or stored permanently in the outer shoal, making the shoal larger and consequently causing refraction patterns at the inlet conducive to shoal growth. In an attempt to determine what the ultimate limits of such growth might be, a correlation was made of the sand stored in the outer shoals with the size of the inlet as determined by its tidal prism P and the amount of wave activity acting on its outer shoal as determined by a wave climatology parameter dependent on the^ energy flux at the 248 249 inlet. Details of . the -sand voluiae calculatiojis are dis- cussed elsewhere (133), and the estimates of tidal prisms and wave parameters as follows . Tidal prism measurements came from References (134, 135, 136, 137, and 138). In most all of the cases the tidal prism wa;s either measured from current data taken at the throat of the inlet or by the "cubature method." Jarrett (13 7) discusses the cubature m-ethod in d-etail . In most all cases the data when the prism was measured corresponded to the survey data from which the estimate of outer bar sand volume was made. Wave heights and wave periods available were average wave heights from wave gages in the nearshore zone (15 to 20 feet below MLW) from the Coastal Engineering Research Center wave gage program. As these wave heights already have the measure of continental shelf slope implicit in them (energy has been dissipated over the shelf up to the wave gage depth) , the basic measure of wave energy used to 2 7 separate energy environments was the parameter H T~ (wave 2 2 height X wave period ). On mildly/ exposed, moderately exposed, and highly exposed coastlines, this parameter was arbitrarily chosen to range from 0-30, 30-300, >300 respectively. This classification lum.ps the South Carolina, Texas, and lower Gulf Coast of Florida inlets into the mildly exposed coast range; the East Coast, and the Panhandle of Florida (Gulf Coast) inlets into the moderately exposed coast range; and the Pacific Coast inlets into the highly 250 exposed coast range. Cojrjrelations were made for three coastal energy level groupings and for all inlets combined using an equation: ^ = aP^ (7.1) where ^ = volume of sand stored in the outer bar/ shoal of the inlet (in cubic yards of immersed sand) , P = tidal prism of inlet (in cubic feet), a,b = correlation coefficients. Linear logarithmic regression was used to obtain b for the case of highly, moderately, and mildly exposed inlets for the case of inlets tabulated in (9). For these four cases the coefficient b equals: Highly Exposed Inlets b = 1.23 Moderately Exposed Inlets b - 1.08 iMildly Exposed Inlets b = 1.24 All Inlets b = 1.26 As there was no significant difference in the exponential correlation coefficients, the value b = 1.2 3 corresponding to the high energy coast (Highly exposed) inlets was used for the correlations with all inlet groupings. The justifi- cation for this somewhat arbitrary fixing of parameters was that a minimum of scatter existed in the correlation of the Pacific Coast Inlets (Figure 7.1), The minimum scatter in this plot over two orders of magnitude is somewhat surprising in view of the many parameters which could be of importance 251 ure 7.1. Tidal Prism - Outer Bar Storage Relationship for Highly Exposed Coasts 252 in inlet outer bar shoaling such as inlet history, avail- able longshore energy flux, and physiography of the inlet- coastal location. Some reasons for some of the scatter seen are as follows : -Inlet history. Should the inlet close or the tidal prism be reduced drastically (due to m.odif icat ions of the inlet inner bay system) much of this material would be driven back to the beaches. The author has noted this occurrence in two occasions on both the East Coast and Gulf Coast of Florida. In the case of the Pacific Coast inlets studied, all of the inlets have been open over recorded history. -Physiography must plan an important part also. The author has noted that inlets i which are estuarine (i.e., estuaries), have significantly smaller inner shoals most likely due to the predominance of the ebb flow during landward flooding periods . The Pacific Coast inlets used for the highly exposed coast correlation are alike in that j they are estuary systems , 253 Thus, the Pacific Coast -inlets ajre .physio graphically similar, historically open, and have similar (in a gross sense) longshore energy flux levels; and, therefore, should experience less scatter. Using the exponential correlation coefficient b = 1.23, analysis was made to determine the correlation coefficient a and the corresponding volume-prism relationship for the three ■ groupings of inlets and for the inlets combined. The corresponding equations are shown below: High exposed coasts (7 inlets) = 8. 7 X lO'^p^-^^ (7.2) Moderately exposed coasts (18 inlets) ^ = 10 . 5 X 10"'^ (7.3) P Mildly exposed coasts (15 inlets) ¥ = 13. 8 X 10~^p^' (7.4) All inlets (44 inlets) ^ = 10.7 X 10~^p^-^^ (7.5) The plots of the prism-outer bar storage volume for the various inlet groupings are given in Figures 7.1, 7.2, 7.3, and 7.4. As inlet channel cross-sectional area shows a definite correlation with tidal prism (134, 135) and is an easier quantity to measure than tidal prism, a correlation was also made with the available data for bar volume — channel cross- sectional area relationships. Correlations were made for the three coastal energy level groupings and for all inlets combined using an equation: 254 a rr. < >- 1000 - 500 - 200 - SAN FRANCISCO 9*Y, CALIF. MOBILE BAY, ALA . -r/. V_ COLUMBIA RIVER, aRP/-~ T" NORTH EDISTO RIVER < 03 X. (Jj I— O <5 '00 -ST. AUGUSTINE INLET, FLA.-f- ST. JOHN'S RIVER ENT. FLA.' 50 - GRA/'S HAR30R, WASH. + CA SRANOe INLET, FLA. ST. Jf'ARY'S RIVER ENTRANCE , FLA. GALVESTON , TEXAS 3TQ<Q0 INLET, 3. C. NYAH SAY, S.C. SAU SOUND , FLA. J- + PENSACQLA HARBOR, FLA. /+ -aSAU.'^ORT' INLET, N.C. + ^ LITTLE EGG INLET, N.J . -p COOS 3AY, ORE, , ARANSAS PASS, TEXAS/ -i-UMPQUA RIVER, ORE. PASS-A-GRILLE INLET, FLA.— ^n-^j. -r OREGON INLET, N.C HEREFORD INLET, N.J.-— '"^/" ;+ TILLAMOOK SAY, ORE BIG SARASOTA PASS, FLA. 4- y '^GREAT EGG HARBOR INLET, N.J. PONCE OE LEON INLET, FLA. 4 -r CAPTIVA PASS, FLA. SARNEGAT INL^T, N.J.-f- 10 - DUNEDIN PASS, FLA. + 4- LONGBOAT PASS, FLA. GASPARILLA PASS FLA.-..i.jy^3„,(3^;^j|f^£ INLET, N.J. NEW PASS,FLA.+-^ jQ„^'3 PASS, FLA. -> -~ - -^J^ -r EAST PASS, FLA. + NEHALEM RIVER, ORE. -L>-CAR0L1NA BEACH INLET, N.C. INDIAN RIVER INLE^^ QEL.4-+ CLEARWATER INLET, FLA. 2 - REDflSH PASS, rl LU 5 _J O > •0 ^ JUPITER INLET, FLA. +/«ENIC£ INLET, FLA. + MIDNIGHT PASS, FLA. 0.5 - + SAKER'S HAULOVER , FLA. 0.2 - Q I I 111 III I 12 5 10 20 50 100 300 SCO lOOO TIDAL PRISM IN CUeiC FEET (x 10° Figure 7.2. . Tidal Prism - Outer Bar Storage Relationship for Inlets on Sandy Coasts 255 0.3 ' ' 111 I t I , 12 5 10 20 50 100 200 500 1000 TIDAL PRISM IN CUSIC rEET(xlO^) Figure 7.3. Tidal Prism - Outer Bar Storage Relationship for Moderately Exposed Coasts 256 ure 7.4. Tidal Prism - Outer Bar Storage Relationship for Mildly Exposed Coasts 257 ¥ = a'A ^' (7.6) c where = volume of sand stored in outer bar/ shoal as before, A = inlet channel, cross-section area at throat (in square feet), a',b' = correlation coefficients Linear logarithmic regression was again used to obtain the coefficient b' for all cases and is tabulated below: Highly Exposed Inlets b' = 1.28 Moderately Exposed Inlets b' = 1.23 Mildly Exposed Inlets b' ~ 1.28 The value b' =1.2 8 corresponding to the highly exposed and mildly exposed inlets was used for the correlations with all inlet groupings, and analysis was made for the coefficient a'. The corresponding equations for all inlet groupings are shown below: Highly exposed coasts (7 inlets) V = 33.1A ^'^^ (7.7) c Moderately exposed coast (18 inlets) 1 9 P V- = 40 . 7A ^ (7.8) c Mildly exposed coasts (15 inlets) ^ = 45. 7A -'^^ (7.9) c The plots of the cross sectional area-outer bar storage volume for the various groupings are not shown, but prove to have considerably less scatter than the tidal prism-outer 258 bj3.r storage volume plots. Under a given set of conditions, hence, both the volume-prism and volume-cross section rela- tionships should be considered when obtaining an estimate of the sand storage capacity of an outer inlet bar system. A nuFiber of parameters other than tidal prism (or cross-sectional area) and wave energy also play a large role in sand trapping on outer bar/shoals. Two important parsm-eter s which have not been -explicitly considered in the present analysis are longshore energy flux which moves the sand to the inlet where the ebb tidal current can deposit on the outer bar, and size distribution of littoral material which limits the ability of the material to movement away from the surf zone. Further research is needed to better define how these parameters control the influence of outer bar sand storage. Further work is also needed on the inner bay or lagoon shoal storage volumes and on the potential of any given inlet to trap sand in its interior shoal system. CHAPTER VIII SUMMARY AND CONCLUSIONS In the preceding chapters a model for sand transport along beaches under the influence of wave action was devel- oped and compared to existing laboratory data and fit to within a constant. The model predicts reasonable values for the integrated (across the surf zone) littoral bed load and suspended load sand transport when compared to existing field and laboratory measured sand transport values. As of yet -though, instrumentation does not exist to perfectly differenti- ate the bed load portion of the sand transport from the sus- pended load portion (139), and thus, the individual bed load and suspended load sand transport models cannot be verified independently . As both field and laboratory measurements of sand trans- port are difficult to make and very costly, few data sets exist which provide all parameters of importance to the cal- culation of sand transport and no data have been taken in which the distribution of sand transport across the surf zone has been measured adequately. Also, field data taken have often left out adequate description of the sediment size dis- tribution and rely mainly on the mean or median grain size, whereas proper measurement should include the fall velocity distribution curve (for suspended load) and the grain size distribution curve (for bed load) . 259 260 Further research should be done on the following sub- jects to improve future sand transport models. --Collection of both field and laboratory measure- ments of sand transport as well as forcing functions (wave parameters) and sediment parameters across the surf zone. --Collection of data to define lateral "mixing" parameter; necessary for predictive ability of dis- tribution of longshore current across surf zone. — Concentration of suspended sediment in the surf zone along with random fluctuations of con- centration and velocity to better define the vertical sediment mixing parameter. --Development of better instrumentation for use in the laboratory and field in surf zones, especially for measuring instantaneous sediment concentrations . Present instrumentation for use in the surf zone is not capable of differentiating between entrained air bubbles and sediment (139), The present model was applied to the prediction of sand transport along the Florida Coast in Chapter V using existing offshore ship wave data. Computations were made by means of computer model which takes into account wave height modification due to energy dissipation by bottom friction. A com.parison is made to sand transport values estimated from dredging records at inlets which provides reasonable agreement in most areas. By a means of data 261 presentation called the "littoral drift rose" based on coastline orientation, the ability to postulate shoreline stability and instability is given. Using thei "littoral drift rose" concept,:" a rational explanation for the existence of natural large cusp features (instability features) oc- curring along elongated bays is presented- In Chapter VI a heuristic model for the static -equilibrium (no longshop.e . sand transport) shape of shorelines sheltered by non-erodable materials (headlands, reefs, etc.) is presented. The model is compared to existing shoreline shapes in the lee of reefs and shoals and is found to provide reasonable results inrjnany areas. Using the developed equation for sand transport, a dynamic model of shoreline change is also postulated. The sand transport model equa- tion is coupled with a volumetric continuity (of sand) equa- tion and reduced to the "heat flux" form of partial differential equations which is then solved for various coastal structure (jetty, beach nourishment, etc.) cases. Chapter VII presents the results of a regression anal- ysis for the volume of sand stored in the outer bars of tidal inlets with the results in the form of equations which involve both inlet tidal prism and wave activity present offshore of the inlet. Results of the analysis allow for equilibrium calculation of inlet outer shoal volume in the event an inlet is to be cut in a barrier island. REFERENCES 1. U.S. Army Corps of Engineers, National Shoreline Study , Washington, D.C., 1971. 2. Hicks, Steacy D. , "Trends and Variability of Yearly Mean Sea Level 1893-1971," NOAA Technical Memo. No. 12 , Rockville, Maryland, March 197 3. 3. Bruun, P., "Sea-Level Rise as a Cause of Shore Erosion," Journal of Waterways, Harbors, and Coastal Engineering , ASCE, Vol. 88, No. 117, May 1962, pp. 183-217. , 4. 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No . 28, Washington, D. C, 1969 . 268 70. Inraan, D. L. , "Wave-Generated Ripples in Nearshore Sand," Beach Ero.si.on Eoard Tec]mi£:al Memo. No. 100 , Washington, D.C., 1957. 71. Das, M. M. , "Longshore Sediment Transport Rates: A Compilation of Data," Coastal Engineering Research Center Miscellaneous Paper No. 1-17 , Washington, D.C., 1971. 72. Krumbien, W. C, "Shore Currents and Sand Movement on a Model Beach," Beach Erosion Board Technical Memo. No . _7, U.S. Army Corps of Engineers, Washington, D.C., 19 44. 73. Saville, T. , "Model Study of Sand Transport Along an Infinitely Long Straight Beach," Transactions of the American Geophysical Union , Vol. 31, No. 4, 1950, pp. 555-56 5. 74. Savage, R. P., "Laboratory Determination of Littoral Transport Rates," Journal of VJaterways , Harbors , and Coastal Engineering , ASCE, Vol. 88, No. WW2 , 1962, pp. 69-92. 75. Ingle, J. 0., Jr., The Movement of Beach Sand , First Edition, Elsevier Publishing Company, Amsterdam, 1966. 76. Galvin, C. 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H. , "Pre- dicting Changes in the Plan Shape of Beaches," Proceed - ings of the 13th Conference on Coastal Engineering, ASCE Vol. 2, Washington, D.C., 1972, pp. 1321-1329, 269 82. Barcelo, J. P., "Experimental Study of the Hydraulic Behavior of Groyne Systems," Memoria No. 35 0 , Labotorio Nacional de Engenharia Civil, Lisboa, Portugal , 1969. 83. Bijker, E. W. , "Littoral Drift as Function of Waves and Currents," Delft Hydraulics Laboratory, Publication No . : 5 8 , Delft, Netherlands, 19 70 . 84. Beach Erosion Board, Interim Report , U.S. Army Corps of Engineers, Washington, D.C., 19 33. 85. Zenkovich, V. P., "Fluorescent Substances as Tracers for Studying the Movement of Sand on the Sea Bed; Experiments Conducted in the U.S.S.R.," Dock and Harbour Authority , Vol. 40, 1960, pp. 280-283. 86. Bruun, P., and Battjes, J. A., "Tidal 'inlets and Lit- toral Drift," 10th International Association for Hy - draulic Research Congress , Vol. 1, London, 196 3, pp. 123-130 . 87. Watts, G. M. , "A Study of Sand Movement at South Lake Worth Inlet Florida," Beach Erosion Board Technical Memo . No ■ 4 2 , Washington, D.C., 1953. 88. Caldwell, J. M. , "Wave Action and Sand Movement Near Anaheim Bay, California," Beach Erosion Technical Memo . No . 6 8 , Washington, D.C., 1956. 89. Bruno, R, 0., and Gable, C. G., "Longshore Transport at a Total Littoral Barrier , " Proceedings of the 16th Conference on Coastal Engineering , ASCE, Vol. 1, Hamburg, Germany ,' Aug . 1978, pp. 128-145. 90. Moore, G. W., and Cole, J. Y., "Coastal Processes in the Vicinity of Cape Thompson, Alaska, Geologic Investiga- tions in Support:-'Of Project Chariot in the Vicinity of Cape Thompson; Northwestern Alaska," Preliminary Report, U.S.G.S. Trace Elements Investigation Report 753 , Washington, D.C., 1960. 91. Longuet-Higgins , M. S., and Stewart, R.W., "Radiation Stress in Water Waves, a Physical Discussion with Ap- plication," De ep-Sea Research, Vol. 11, No. 4, 1964, pp. 529-563. 92. Longuet-Higgins, M. S., and Stewart, R.W., "A Note on Wave Set-Up," Journal of Marine Research , Vol. 21, 1963, pp. 44-100. 93. Le Mehaute", B., and Webb, L.M., "Periodic Gravity Waves Over a Gentle Slope at a Third Order of Approximation," Proceedings of the 9th Conference on Coastal Engineering , ASCE, 1964, -pp. 23-40. 270 94. Phillips, 0. M. 5 The Dynamics of the Upper Ocean , First Edition, Unav.epSui±y Press., .Cajnbrxdge, Mass., 1956, pp. 22-70. 95. Whitham, G. B., "Mass Momentum and Energy Flux in Water Waves," Journal of Fluid Mechanics , Vol. 12, No. 2, 1962, pp. 135-147. 96. Kinsman, B., Wind Waves , First Edition, Prentice Hall, Englewood Cliffs, N.J., 196 5. 97. Bowen, A. J., Inman, D. L. , and Simmons, V. P., "Wave Set-Down and Set-Up," Journal of Geophysics Research , Vol. 73, No. 8, 1968, pp. 2559-2577. 98. Iv-erson, -H . W., "Wav-es --arid- Bi?eakers In Shoaling Water," Proceedings of the 3rd Conference on Coastal Engi- neering , ASCE, 1953, pp. 1-12. 99. Larras, J., "Experimental Research on the Breaking of Waves," Ann. Fonts Chaussees , Vol. 12 2, Grenoble, France, 19 52, pp. 5 2 5-542. 100. Ippen, A. T., and Kulin, G., "Shoaling and Breaking Characteristics of the Solitary Wave," Massachusetts Institute of Technology Hydrodynamics Laboratory Report 15 , Cambridge, Mass., 195 5. 101. Eagleson, P. S., "Properties of Shaaling Waves by Theory and Experiment , " Transactions of the American Geophysical Union , Vol. 37, 1956, pp. 565-572. 102. McCowan, J., "On the Highest Wave of Permanent Type," Phil. Mag . , Vol. 39, 1894, pp. 351-359. 10 3. Davies, T. V., "Symmetrical, Finite, Amplitude Gravity Waves," Gravity Waves , NBS Circular 521, Chapter 9, 1952, pp. 55-60. 104. Long, R. R. , "Solitary Waves in One- and Two-Fluid Systems," Tellus , Vol. 8, 1956, pp. 460-471. 10 5. Laitone, E. V., "Limiting Conditions for Cnoidal and Stokes Waves," Journal of Geophysics Research , Vol. 67, No. 4, April 1962, pp. 1555-1564. 106. Dean, R. G., "Breaking Wave Criteria: A Study Employing a Numerical Wave Theory," Proceedings of the 11th Conference on Coastal Engineering , ASCE, Vol. 2, Washington, D.C., 1968, pp. 1053-1078. 107. Stoker, J. J., Water Waves , First Edition, Interscience ■ Publishers , New York, 196 8. 271 10 8. Munk, W. H. , "The Solitary Wave Theory and Its Applica- tion to Surf Problems , " ATinual of th-e New York Academic Science , Vol. 51, No. 3, 1949, pp. 376-424. 109. Thornton, E. B., "Review of Longshore Currents and Theories , " Proceedings of the National Sediment Trans - port Study Conference , Newark, Del., Oct. 1976, pp - 21-33 . 110. Einstein, H. A,, "Bed Load Function for Sediment Trans- portation in Open Channel Flows," U.S.D.A., S.C.S., Technical Bulletin 1026 , Washington, D.C., 1950. 111. Personal Conversatio n with R. G. Dean, J. Basillie and R. Bruno. 112. Watts, G. M., "Development and Field Tests of a Sampler for Suspended Sediment in Wave Action," Beach Erosion Board Technical Memo. No. 34 , U.S. Army Corps of Engineers 5 Washington, D . C . , 19 5 3 . 113. Fukushima, H., and Mizoguchi, Y., "Field Investigation of Suspended Littoral Drift," Coastal Engineering in Japan , Vol. 1, 1958, pp. 131-134. 114. Fairchild, J. C. , "Longshore Transport of Suspended Sediment," Proceedings of the 13th Conference on Coastal Engineering, ASCE, Vol. 2, Vancouver, B.C., 1972, pp. 1279-1305. 115. Hom-ma, M. , and Horikawa, K., "Suspended Sediment Due to Wave Action," Proceedings of the 8th Conference on Coastal Engineering , ASCE, Mexico City, 1962, pp. 753-771. 116. Shinohara, K., et al., "Sand Transport Along a Model Sandy Beach by Wave Action," Coastal Engineering in Japan , Vol. 1, 1958, pp. 211-223. 117. Hattori, M., "A Further Investigation of the Distribu- tion of Suspended Sediment Due to Standing Waves," Coastal Engineering in Japan , Vol. 14, 19 71, pp. 3 2-56". 118. Kennedy, J. F. , and Locher, F. A., "Sediment Suspension by Water Waves," Institute of Hydraulic Research Report ' 15_, University of Iowa, Iowa City, 19 75 . 119. Fairchild, J. C, "Suspended Sediment in the Littoral Zone at Ventnor, New Jersey, and Nags Head, North Carolina," Coastal Engineering Research Center Tech - nical Paper TI-5 , Washington, D.C., 1977. 272 120. Kana, T. W. , "Suspended Pediment Transport at Price Inlet, S.C.," Coastal Sediments '77 , ASCE, Charlestown, South Carolina, Nov. 1977, pp. 798-821. 121. Bagnold, R. A., "Experiments on a Gravity-Free Dis- persion of Large Solid Spheres in a Newtonian Fluid Under Shear," Royal Society (London) Proceedings , Vol. 225, 1954, pp. 49-51. 122. U.S. Naval Weather Service Command, "Summary of Synoptic Meterological Observations (SSMO) for North American Coastal Marine Areas," Vols. 4 and 5, Ashville, N.C., 19 72 . 123. Le Mehaute, B., and Koh, R.C.Y., "On the Breaking of Waves Arriving at an Angle to the Shore," Journal of Hydraulic Research , Vol. 5, No. 1, 1967, pp. 67-88. 124. Walton, T. L. , Jr., and Dean, R. G., "The Use of Outer Bars of Inlets as Sources of Beach Nourishment Material," Shore and Beach Magazine, Vol. 44, No. 2, July 1976, pp. 13-19. 125. Pelnard-Considere , R. , "Essai de Theorie de I'Evolution des Formes de Rivate en Plages de Sable et de Galets , " 4th Journees de 1 ' Hydraulique , Les Energies de la Mar, Question III, Rapport No. 1, 1956. 126. Yasso, W. E., "Plain Geometry of Headland-Bay Beaches," Journal of Geology , Vol. 73,1965, pp. 702-714. 12 7. Silvester, R. , "Growth of Cenulate Shaped Bays to Equilibrium," Journal of Waterways, Harbors, and Coastal Engineering , ASCE, Vol. 96, No. WW2 , 1970, pp. 275-287. 12 8. Johnson, D. W. , Shore Processes and Shoreline Develop - ment , First Edition, Hafner Publishing Company, New York, 19 72. 129. Mehta, A. J., and Christensen, B. A., "Incipient Motion of Shells as Dredge Material Under Turbulent Flows , " Seminar XVII Congress of International Associa - tion for Hydraulic Research, Baden-Baden, Germany, Aug. 197 7. 130. Wiegal, R. L., Oceano^raphic Engineering , First Edition, Prentice-Hall Publishing Co., New York, 1960. 131. Carslaw, H. S., and Jaeger, J. C, Conduction of Heat in Solids, First Edition, Clarendon Press, Oxford, 19 59. 273 132. Abramowitz, M. , and Stegun, I. A., Handbook of Mathematical Functions , First Edition, Dover Publica- tions, Inc., New York, 19 72. 133. Dean-,- R. G., and Walton, T. L. , "Chapter I, Geology of Estuaries," Proceedings of the Second International Estuarine Research Conference , Academic Press, 1973. 134. O'Brien, M. P., "Estuary Tidal Prisms Related to Entrance Areas," Civil Engineering, Vol. 1, No. 8, 1931, pp. 738-739. 135. O'Brien, M. P., "Equilibrium Flow Areas of Inlets on Sandy Coasts," Journal of Waterways, Harbors, and Coastal Engineering, ASCE, Vol. 40 No. WWl , 196 9, pp. 823-839. 135. Johnson, J. W. , "Tidal Inlets on the California, Oregon, and Washington Coast," University of California, Hydraulic Engineering Laboratory Report HEL 24-12 , Berkeley, California, 19 72. 137. Jarrett, J. T. , "Tidal Prism-Inlet Area Relationships," U.S. Army Waterways Experiment Station, Vicksburg, Mississippi, 1974 (unpublished). 138. Bruun, P., and Gerritsen, F., Stability of Coastal Inlets, Vol. 1 and 2, North Holland Publishing Company, Amsterdam:, 19 6 6 139 . Sea Grant Nearshore Sediment Transport Study Workshop on Instrumentation for Nearshore Processes , Lajolla, California, June 19 77. APPENDICES APPENDIX A DERIVATION OF LONGSHORE ENERGY FLUX PER UNIT LENGTH SHORELINE The derivation for P,^ proceeds as follows: Assume a coast with contours that are parallel to a straight shore- line (Figure A-1) . Waves approaching this coast are assumed to -bfi. -described by linear..- .smaJJ. amplitude theory. In gen- eral, a wave crest that makes an angle with the shoreline when in deepwater will refract to make an angle a at some shallower depth (Figure A-1) , where a is related to by Snell's law. In what follows, the subscript, o, refers to deepwater conditions. The path of a wave passing through point i is shown on Figure A-1 as the dashed orthogonal labeled "wave path." The flux of energy between orthogonals in the direction of wave travel at point i is given by P. = C 1 ^ ^ (A-1) = (n CE)^£^, where C is the wave group velocity, C is the wave phase S velocity, n = C /C, E" is the energy density, the total average energy per unit area of sea surface, r = CA-2) 275 Orthogonal 2 Wave Crest R3int ■ Wave Ray u Bottom Contours Beach Figure A-1. Definitions- for Conservation of Energy Flux for Shoaling Wave 277 where y is the weight density of water (64.0 lbs/ft for sea water) and H is the wave height. (This wave height is the height of a uniform periodic wave). Small amplitude theory is assumed, energy is assumed constant between orthogonals in this approach and does not spread laterally across wave orthogonals. Therefore, the energy flux in the direction of wave travel must remain coiistant.-. beiween . ,Qrthogoxia2^-,.,-±ha.t . j-s, betweezL deepwater and point i, P = P. = constant, (A-3) o 1 where is defined by equation A-1. However, the longshore component of P^, designated P,^, where P^ = P^ sin a (A-14) does change along the wave path, since a changes due to refraction while P^ is constant by equation A-3. The sub- script i in this appendix indicates any point on the wave path, including deepwater, where small amplitude theory holds . From the geometry of Figure A-1, it is obvious that changes with position on the wave path. However, the dis- tance between adjacent orthogonals, b, measured parallel to the coast does not change. Therefore, at any point, i, in the wave path Jl. = b cos a. (A-5) 278 From equations A-4 and A-5, this longshore component of energy flux can be written P„ = (E C b cos a) sin a (A- 6) For the straight parallel contours assumed, the distance b is arbitrary. Arbitrarily setting b = 1 foot (or a unit length in the system of measurement used), one finds that the longshore power per unit length of shoreline is : = E C sin a cos a or using a trigonometric identity P„ = E C sin2a ^ 2" g (A-7) (A- 8) At breaking, where the longshore wave energy dissipated shoreward of breaking has been assumed to be the available energy for transporting sand in the longshore direction (ttC ), sin 2a, 2 g b' b or £b at breaking ^Ih = -16- ^gb ^% (A- 9) (A-10) APPENDIX B LONGSHORE SAND TRANSPORT MODEL DATA SUSPENDED SAND CONCENTRATION DATA 280 PL. 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Kana Data. SERIES STA H(ft.) T(sec. ) D(ft. ) Y(ft.) C(ppt) 10107 B02 1.15 10 1.81 .33 .141 .99 .039 1.98 .016 10229 C+l 1.4-8 9.5 2.38 .33 .306 .99 .097 1.98 .038 10343 PIl 1.48 8.5 2.70 .33 .323 .99 .175 1.98 .083 10564 PI9 1.80 12.0 2.86 .33 .132 .99 .081 1.98 .028 10668 CAl 1.96 11.0 3.11 .33 .259 .99 .192 1.98 .089 20109 BIl 1.96 8.0 2,94 .33 1.241 .99 .789 1.98 .351 20547 CAl 3.61 9.5 5.08 .33 1.244 .99 .702 1.98 .647 3.28 .361 20658 BU2 2.95 8.5 4.77 .33 0.499 .99 0.278 1.98 0.194 3.28 0.113 20767 PI9 2,30 7.0 3.61 .33 .258 .99 .137 1.98 .059 30216 PIl 1,80 11.0 2.70 .33 .595 .99 .301 1,98 .218 30218 PIl 1.80 11.0 2.86 .33 .376 .99 .221 1.98 .081 30442 BU2 1.97 11.5 5.20 .33 .707 .99 .280 1,98 .073 40214 PI9 1.80 9.5 4.03 .33 .149 .99 .064 1.98 .033 285 Table B-5. Fairchild Data. STA. H(ft.) T(sec) D(ft.) Y(ft. ) C(ppt) 234-N 2.18 6.6 4-. 70 0.20 0.328 2.18 6.6 ^.72 0,20 0.461 1.82 9.2 4-. 74 0.39 0.4-52 1.82 -9.2 4.76 0.39 0.457 2.00 8.0 4.78 0.39 0.292 2.00 8.0 4.77 0.39 0.514 1.73 6.9 4.75 0.59 0.183 1.73 6.9 4.74 0.59 0.777 1.73 ..6,-9 4.72 0.78 0.105 4.72 0.78 0.320 2.00 8.7 4.61 0.98 0.463 2.00 8.7 4.59 0.98 0.185 2.00 8.7 4.52 1.18 0.235 2.00 8.7 4.48 1.18 0.351 1.92 7.2 4.30 . 1.37 0.253 1.92 7.2 4.28 0.20 1.774 1.71 6.8 4.11 0.20 1.438 2.12 6,7 3.88 0.39 1.589 2.12 6.7 3.87 0.39 0.951 325-S 2.40 4.4 3.48 0.19 1.353 2.40 4,4 3.49 0.19 1.429 2.72 6.7 3.52 0.39 1.036 2.71 6.7 3.66 0.39 0.783 4.18. 5.0 4.15 0.58 1.552 4.18' 5.0 4.19 0.58 1.183 4.18 5.0 4.25 0.77 0.915 3.54 5.3 4.31 0.77 0.915 3.54 5.3 4.45 0.97 0.818 3.51 5.1 4.47 0.97 0.689 325-S 3.51 5.1 4.57 1.45 0.515 3.51 5.1 4.59 1.45 0.736 4.67 1.93 0.876 4.69 1.93 0.875 2.95 5.4 4.32 2.42 0.481 2.95 5.4 4.83 2.42 0.409 2.99 5.5 4.86 2.90 0.392 2.99 5.5 4.87 2.90 0.547 3.00 5.4 4.93 0.19 3.343 3.00 5.4 4.94 0.19 3.228 286 Table B-6. Continued. STA. H{ft.) T(sec) D(ft.) Y(ft.) C(ppt) 354-N 2.08 5.5 4,80 0.59 1.000 2.08 5.5 4.78 0.59 0.891+ 1.86 5,4. 4.59 0.78 1.125 1,86 5.4 4.57 0.78 0.933 1.86 5.4 4,52 0.98 0.927 1.86 5.k 4.49 0.98 0.584 1.65 5.5 4.23 0.20 1.502 1.55 5.5 4.21 0.20 1.285 1.40 5,5 4.07 0.39 1.178 1.40 5.5 4.05 0.39 1.441 354-N 2.20 5,8 3.91 0.59 0.681 2.20 5.8 3.86 0.59 1.568 1.65 5.8 3.75 0.78 0.703 1.65 5.8 3.74 0.78 0.425 1.65 5.8 3.62 0.20 1.101 1.64 6,0 3.60 0.20 1.714 1.49 9.,9 3.53 0.39 0.692 1.49 9.9 3.51 0.39 1.613 1,1+9 9.9 3.50 0.39 1.438 1.72 5.0 3.47 0.39 1.200 1.72 5.0 3.41 0.39 0.728 326-N 1.85 9.1 2.33 0.20 0.456 1,85 9.1 2.34 0.20 2.181 1.85 ■ 9.1 2.43 0.39 1.454 1.75 8.8 2.44 0.39 0.923 1.75 8.8 2.55 0.59 0.710 1,70 9.1 2.58 0.59 0.865 1.65 9.3 2.66 0.78 0.765 1.65 9.3 2,72 0.78 0.147 1.82 9.3 2.76 0.98 0.288 1.82 9.3 2.77 0.98 0.537 326-N 1.90 9.2 3.51 0.59 0.368 1.90 9.2 3.54 0.59 0.550 1,90 9.2 3.60 0.78 0.253 1.90 9.2 3.64 0.78 0.515 1.77 9.3 3.73 0.98 0.158 1.77 9.3 3.74 0.98 0.271 1.53 10.0 3.57 0.39 1.473 1.53 9.9 3,44 0,39 1.992 1.45 9.2 3.34 0.39 1.548 1.45 9.2 3.33 0.39 0.993 287 Table B-5. Continued. STA. H(ft.) T(sec) D(ft.) Y{ft.) C(ppt) 326-N 1.68 9.9 3.87 1.47 0.145 1.68 9.9 3.87 ■ 1.47 0.148 1.59 8.8 3.91 1.96 0.033 1,59 9.8 3.92 1.96 0.294 1.57 9.3 3.94 0.39 1.206 1.57 9.3 3.93 0.39 0.951 1.57 9.3 3.90 0.39 0.588 1.57 9.3 3.89 0.39 0.593 3i+0-N 0.96 10.0 2.76 0.19 0.180 0.96 10.0 2.75 0.19 0.115 1.09 10.1 2.77 0.39 0,081 1.09 10.1 2,77 0.39 0.013 1.09 10.1 2.77 0.58 0.052 1.01 9.6 2.77 0.58 0,113 1.01 . 9.6 2.79 0.77 1.01 9.6 ■ 2.84 0.77 0.031 1.02 9.6 ' 2.85 0.97 0.085 1.02 9.5 2.87 0.97 340-N 1.02 9.6 2.94 1.45 0.81 9.8 2.94 1.45 0.81 9.8 3.03 1.93 0.82 9.4 3.05 1.93 0.82 9.4 3.13 2.42 0.82' 9.4 3.14 2.42 325-N 0.67 10.1 2.47 0.58 0.100 0.67 10.1 2.45 0.58 0.091 0.57 10.1 2.44 0.78 0.030 0.70 10.2 2.43 0.78 0,034 0.70 10-2 2.43 0.97 0.048 2.43 0.97 0.078 2.44 1.46 2.45 1.46 0.52 10.3 2,47 1.95 0.52 10.3 2.49 1.95 0.046 0.55 10,0 2.54 0.20 0.311 0.55 10.0 2.55 0.19 0.347 0.56 10,5 2.54 0.39 0.125 0.55 10.5 2.65 0.39 0.065 1.15 10.1 2.83 0,58 0.039 1.16 10.1 2.85 0.58 0.030 288 Table B-6. Continued. STA. H(ft.) T(sec) . D(ft.) Y(ft.) C(ppt) 258-N l.i+U 10.3 2.54 0.19 1.274 1.4-4 10,3 2.55 0.19 2.380 1.20 10.1 2.53 0.39 0.283 1.20 10.1 2.54 0.39 0.498 0.74 10.4 2.71 0.58 0.165 0.74 10.4 2.71 0.58 0.158 1.43 9.9 2.77 0.77 0.076 1.43 9.9 2.77 0.77 0.174 1.07 11.8 2.76 0.97 0.078 1.07 11.8 2.76 0.97 0.112 1.30 5.3 2.74 1.45 0.094 1.30 5,3 2.73 1.45 0.102 1.30 5.3 2.70 0.19 1.712 0.99 10.1 2.69 0.19 0.751 366-N . 4.30 1.45 0.083 4.26 1.93 0.099 4.26 1.93 0.047 1.87 8.2 4.24 2.42 1.073 1,87 8,2 4.22 2.42 0.087 2.12 8.0 4.21 2.90 0.148 2.12 8,0 4.21 2.90 0.097 2.06 8.5 4.20 0.19 1.474 2.06 8.5 4.20 0.19 1.477 1.92 8.1 4.21 0.39 0.207 1.92 ■ 8.1 4.22 0.39 0.015 2,12 7.5 4.23 0.58 2.12 7.5 4.26 0.58 0.040 APPENDIX C ANALYSIS OF SSMO WAVE HEIGHT, PERIOD, AND DIRECTION RANGES The purpose of this appendix is to describe the manner in which the groupings of wave data listed in the SSMO volumes were handled for computations of longshore energy flux. Wave Height For the SSMO data, a representative value of H^ must be chosen for each interval of wave heights contained in SSMO Tables 18 and 19. Since energy is a function of wave height squared (in linear theory), a representative value of H^ for a given range of H^ values should be based on the mean square root value of the wave height over the range. Consider the probability of occurrence of a wave with specific height H as equal to p(H) in the range H^ to H2. The energy represented in this band of wave heights is proportional to H;^ the mean value of a representaxiive wave height squared where : J p(H) H^ dH h' = ^ r fH2 p(H) dH ^-1 289 290 Since pCH) is not known,, it J_s .cnnsi dered uniform, which is reasonable if the wave height range < H < H2 is small. The equation then becomes: H^dH 1(H2 - hJ) 3(H2 - H^) (II-2) dH Taking the square root of this value, 1/2 H 1(H2 - hJ) _3(H2 - H-^)_ (II-3) Using Equation (II-3), representative values of were found for the corresponding ranges of given in SSMO data, and are summarized in Table C-1. Wave Period Representative values of T were assumed to be the average of the SSMO period ranges, and are given in Table C-2. For T > 13.5 seconds a representative value of T = 16 seconds was assumed. Wave Direction Directional observations as recorded in the SSMO volumes are given on eight points of the compass and thus correspond to eight M-5° sectors of the compass. In the computation of the longshore energy flux, the midpoint of the sectors, as 291 Table C-1. Representative Values of Wave Height Used in Computation of Longshore Energy Flux. Actual SSMO Range of Heights Height Used in SSMO Coded Height (feet) Computation (feet) <1 1-2 .82 - 0.82 2.46 0.47 1.71 3-1+ 2.46 4.10 3.31 5-5 4.10 5.74 4.94 7 5.74- 7.38 5.58 8-9 7.38 9.04 8.22 10-11 9.04 10.70 9.85 12 10.70 12.30 11.49 13-16 12.30 15.50 13.98 17-19 15.60 18.90 17.25 20-22 18.90 22.15 20.53 23-25 22.15 22.43 23.81 Table C-2. Representative Values of Wave Period Used in Computation of Longshore Energy Flux. SSMO Coded Period Used in Computation Period Actual SSMO Range of Periods of Longshore '- <5 0 < T < 5,5 3.0 6-7 5.5 7.5 6.5 8-9 7.5 9.5 8.5 10-11 9.5 11.5 10.5 12-13 11.5 13.5 12.5 >13 13.5 CO 16. 292 givexi in the SSMO data .by ..tlie.- eight pn±Qts of the compass, were used as the representative values of 6 for direction of wave approach. When a representative wave having a given frequency was parallel to the coastline, the corresponding sector of waves was divided into two parts , one being de- leted from the computation and the other approaching the coastline from the midpoint of its half sector with the coTTresponding frequency halv-ed- (see Figure C-26). Wave data in octants with midpoints in the off-shore direction (> e + SQ°) for a given coastline orientation have been n — deleted from the drift computations. In the SSMO data it was ascertained that a considerable number of the original observations were taken on the 3 6 points of the compass, and, when reduced to the eight points of the compass in the SSMO tables, a skew of the wave direc- tion was introduced. This skew amounts to a ten degree shift clockwise, and has been accounted for in the results of the littoral drift computations. 293 Wove data offer modification Figure C-26. 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E > (13 < 1— M- 1 O O O 'r- •1— 4-> +J fO •r— S- n3 > o CM OJ S-- 315 u ta cQ -a i- ^ o +J <— •r- U. 5 -M -fJ Z c O 1—1 I— <u (13 O S_ -r- O C +j •+J > _1 o n3 I/) O «3 I— a. r— id Z3 O C t/1 C rt3 rO CD CD (O CT S- oj en > <£. I M- £= O O SZ -M O fO ■r- -t-J -l-J C fC O) •r- "r- 1. S- <0 CD I •I— Ll_ 316 111 I ^ 317 •s *i 'i I 318 (O -r- CQ O J= U- 3 >5 03 +J CQ •I- (/) %~ o Q " — S- 1— <T3 « o o c +-> <a 4-5 C/1 _1 o r— ro 4J +J CU O I— c: c C rtJ <a o s- o <D CQ > < 1 ^- c o o •r- O 13 •r- 4-> +-> C s- s- IT3 O > CO CM I O <u %. Z3 319 ? ^ B 320 o ra CD CO ^ i- ■r" o 3 LL. +-> 4- •r- o S- s= d s o m ai s- o <u 4J Q. +-> •I — o _J o r— n3 +-) O 1— Cl. (/) Z3 C •r" C 05 •a: 05 •r" 0) 3 cn ta 1 s- <v c > o 'r- 4- ra o +-> c: <u o ■r" s- +J O- tO •r" s- > Ln I o cn BIOGRAPHICAL SKETCH Todd L. Walton Jr. was born on March 17, 194 7, in Dayton, Ohio. He graduated from Belmont High School, Dayton, Ohio, and then entered the University of Cincinnati, in Cincinnati, Ohio. While in this cooperative college, he worked as an engineer with the Baltimore and Ohio . Railroad and the Southwestern Portland Cement Company to pay his way through school. He graduated in June 19 70 with a Bachelor of Science degree (with honors) in Civil Engineering. After undergraduate school he enrolled in a graduate program at the University of Florida and completed a Master's degree in coastal and oceanographic engineering (with high honors) in June 1972. During the years from 1970 to 1972 he also completed a Reserve Officers Training Course.%and was com- missioned as a 2nd Lieutenant in the U.S.A.R. Upon gradua- tion, he fulfilled his military obligation with the U.S. Army Engineers at Ft. Belvoir, Virginia, and joined the faculty at the University of Florida as an assistant engineer.. In the years since graduation he has taught classes in coastal structures and littoral processes, worked in research in the area of coastal hydrodynamics and beach erosion problems, and served as the Coastal Engineering Specialist to the Florida Sea Grant Program, as well as 321 322 serving on a Governor's Task Force (Florida) on Beach Erosion and Inlet Maintenance. He then decided to pursue more graduate studies. "For thou hadst cast me into the deep, in the midst of the seas; and the floods compassed me about; All thy billows and thy waves passed over me" (Holy Bible. Jonah II : 3) . I certify that I have read this study and that in niy opinion it conforms to acceptable standarxis of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. B. A. Christensen, Chairman Professor of Civil Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. T. Y. Chiu>^ Associate Professor of Coastal and Oceanographic Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. J. G. Melville Assistant Professor of Civil Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Daniel P. Spangler Associate Professor of Geology This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate Council, and was accepted as partial ful- fillment of the requirements for the degree of Doctor of Philosophy. June 1979 Dean, College of Engineering Dean, Graduate School